A couple of weeks ago I got home from a 2 hour highway trip and coolant was pouring out of the bottom of the engine so I assumed it was time for yet another (just had a new one put in 3 years ago) water pump on my 2001 Cavalier 2.2. But I am so out of cash that I decided that I would see how I could manage just driving around for a while and adding coolant as it got lower over time. So that’s what I’ve done twice now in the last 20 days. It isn’t cheap at $14 a gallon for DexCool but if I can nurse it along and keep it from over heating (which I have so far), I’ll keep doing it unless or until it gets worse. Anyway, it behaves normally until it starts to get low on coolant and at that time, I know I need to add more by watching how hot the temperature gets BEFORE the thermostat opens. After it does, the temp drops back down to near normal, no matter how low on coolant I seem to be. Someone told me that I should change the thermostat and that could be the cause of a coolant leak. I was puzzled by this. It is the original thermostat at over 16 years and 219K but it it obvious to me that it is opening and closing. So a leak near the water pump - would be a leaking water pump and not a sticking/failing thermostat, right?
I suggest you let the everything completely dry out first. THen add the amount of coolant you need without starting the engine. Once fluid is added start the engine and observe where the water leaks. Could be around the thermostat, a busted or leaking hose or around the water pump.
What you are doing now, simply adding more coolant as you need it is expensive and not good for your engine.
A leaking water pump or thermostat housing is a pretty obvious one. Look to the pulley that drives the water pump. If there is a stream of coolant running down below that pulley, its pretty likely the water pump. Same for the thermostat housing. Both can fail without actually leaking, especially the thermostat causing the coolant to boil over. The boil-over is pretty obvious as well as it is blowing out the overflow tube next to the radiator cap.
If the coolant level is dropping withOUT an obvious leak or boil-over, you have a far more serious problem - the head gasket.
You should be careful about this. Most designs have the temperature sender element poking out into the water jacket. When coolant runs low, it can be surrounded by air pocket and not accurately reporting the engine temp anymore…
Poor maintenance of cooling systems can cause pumps to wear out prematurely, although I have gotten as little as 50K on a rebuilt water pump. Whether the thermostat is leaking or not, never replacing it for 16 years suggests a lack of maintenance. Now, not as a mechanic, but Dex is OK as long as there are no air leaks. Once you have a leak, it gum up the system. I strongly suggest that you get this fixed and the cooling system serviced. Like we used to say, its not a matter of having money its a matter of priorities.
Yow! Have you blown it up, yet? There is such thing as being penny wise and pound (big bucks) foolish!
Tough love… As your new best friend, I say, “Park it until it gets diagnosed and repaired.” Sorry, sometimes even best friends have to speak up. 
Dex-Cool is what I use in all my many GM cars. It’s great coolant, whether it gets air in it or not. About the only ways you can screw it up are leaving it in for 7 or 10 years or adding green coolant to it to top it off.
Go to plan B (No, not the one about adding coolant for a while to get by), the other one, which involves figuring out how to afford to do what’s right and stop the coolant consumption/leak. 
CSA
CSA’s advice above could save you the cost of a much more expensive head gasket job.
As far as what it is that’s leaking, it’s probably the water pump like you speculate. Usually when that happens on my truck it’s very obvious what’s leaking just by looking at it. And if I grab hold of the water pump pulley at 3 and 9 o’clock [engine turned off of course], I can rock it back and forth with little effort, another big clue. There’s probably a small hole on top or bottom of the water pump called the “weep hole”, and if you see any water dripping out of that hole, the water pump is shot. A little dampness inside the hole is ok, but any water dripping out, pump is no good.
Coolant hoses can spring a leak, usually at the connectors. That’s not an uncommon thing. Check for that idea too. Hoses can completely burst too, but I doubt that’s your current problem. The thermostat housing where it attaches to the block or head could leak, but that’s not a common place for large coolant leaks. Small weeping leaks can occur there, but usually not big leaks. The thermostat itself could leak, but that would be an internal leak, not visible. Radiators are a pretty common source of leaks, especially when they reach the 10+ years age. Look for puddles of coolant on the bottom shelf of the radiator.
As posted above, replacing the coolant with fresh every two years can save you a lot of coolant system grief. The coolant tends to get overly acidic otherwise. The rust inhibitors, lubricants, and pH buffers all deteriorate with time.
A couple of things. It isn’t leaking from any hoses or anywhere near the thermostat. I have spent hours looking from the top and the bottom - and cannot see exactly where it is coming from - but it is very close to the water pump and exactly the same location that my last leaking water pump leaked before it was replaced 3 years ago. I had enough money at that time to have it fixed, unlike now.
The leak is EXTREMELY intermittent. It always leaks at least a few drops, sometimes it leaks a lot of drops - and sometimes - it leaks A LOT - I would say about 1/4 gallon all at once. I have tried and tried to determine the variables involved in what causes the major leaks - time driving, highest temperature reached during drive, temperature at time of engine shut-off, etc…and there is no apparent connection for any variable that I can tell, so my assumption is that it is like Russian Roulette and it just depends on what exact point that pump stops at - it must have a leak at one spot in the rotation and if it stops at the wrong spot - I get a gusher.
I have had to add about 1/2 gallon of coolant 3 times now since this started - and right now, I need to do it again because unless I am going a constant 40mph or more without stopping, the temp gets too high for my comfort level. Even a long traffic light is a problem.
So back to my original question: if there is a leak coming from the water pump area - is it or is it not possible that it is related to a faulty thermostat rather than a faulty water pump?
The simplest line of reasoning would suggest that if the leak is coming from around the water pump that the water pump is the problem.
I’m trying to understand why you tie the thermostat together with the water pump? How could a faulty thermostat cause a leak in the water pump? Unless of course the stat caused over-heating and blew a hose or seal or something. So no it is not a thermostat issue but why would you not replace a 16 year old thermostat when you are working on the cooling system? I replaced my 4 year old one as a preventative measure. So the answer is no.
Where are these water pumps coming from?
Some aftermarket parts are junk.
I had a friend back in the day with a 1980 Accord.
He was putting a water pump in it every year.
A lifetime warranty unit from the now defunct Trak Auto chain.
I got him a pump from the dealer.
It cost 2X as much as the aftermarket, but it lasted for many years (until he traded in the car.)
Concur w/ @circuitsmith above , an aftermarket water pump requiring replacement at 3 years isn’t that unusual. I don’t see how a water pump leak could be related to the thermostat. If the thermostat stuck open, that would cause the engine to run cooler than it should, but it wouldn’t affect the water pump. If the thermostat stuck closed that would cause the engine to run hot and overheat, but that wouldn’t usually affect the water pump either. Have you tried rocking the water pump’s pulley, checking for play?
I can understand your ‘between a rock and a hard place’ problem, needing a new water pump but not enough money to do the job. I presume you also understand the posters above who describe expenses you may incur by not replacing the water pump. Like I say, rock and a hard place. One idea, if you’re not sure whether the problem is the water pump or not, hire a shop to give you a diagnosis. That shouldn’t cost much, just for the diagnosis. You don’t have to have them fix it; you can decide how to do that later. Then you’ll at least know for sure what the actual problem is.
From what I can see, an oem pump costs around $180, and it’s around a 2 hour job. So all together you’re looking at a little less than $400 for a new oem water pump. Are you sure the uncertainty of getting where you are going you’ve got now isn’t worth $400 to eliminate from your life?
Be aware that a water pumps that leaks when the engine is operating may not leak when the engine is off. The pump shaft when spinning can allow a leak that closes up when the shaft stops spinning.
You can buy at any parts store a kit with a UV-sensitive dye and a blacklight. Follow the directions, which will basically be to add the dye, take a drive, and check for leak trails with the blacklight. The leak will light up when the UV light hits it.
Use a flashlight and a small mirror and see if you can look at the water pump almost all the way around, behind the belt-driven pulley. You should find the small weep hole (probably on the bottom).
From my experience there will usually be a telltale sign of a leak from that hole, in the form of coolant stain/residue, or a cleaned off trail through the engine grime/dust, indicating that it has been leaking.
Also, from my experience, water pumps sometimes do leak badly when the engine is shut off and the engine to continues to heat, not so much while running.
CSA
I once had a water pump that only leaked on a cold start for 30 seconds or so until it got warm.