Is is still possible for the electric windows and my radio to still work even if my battery is dying?
Yes.
The starter motor draws more electrical current than almost anything else on the car. Many people find that their battery does not have sufficient power to turn the starter motor, even though various accessories on the car still function.
Yes, the battery may be going downhill and not be able to crank he engine, but the windows or lights, which equire a small fraction of the the power, may still work.
Not sure why you are asking the question, but you should have a battery load test done by a good shop. They can tell you how good the battery still is.
Yes, especially if you have a large engine and a large battery. What kind of car is it?
Many auto parts stores will test your battery for free.
I may have a similar problem. Sometimes my starter will do absolutely nothing but everything else works perfectly normally. Could a low battery lead to no cranking at all, rather than slow cranking? The shop just replaced the starter, it is 19 yr old, as a fix for my intermitant starting failures. But now I wonder if it was just the battery I needed to change, not the starter. My impression was that the starter would still try to turn over even with an underpowered battery but perhaps starters have a threshold voltage they need to even try?
Yea, but many auto part stores will do a battery and charging system test for free. We have to guess, but a simple test will tell you the answer most of the time. BTW I would include the charging system in the list of possible problems.
If you would have told us how old the battery was and maybe what make model and year car, we might make better guesses. 
Get the battery tested for free at a local auto parts store and you will have the answer to your question.
I know the battery is bad; the garage told me that after testing it. My question is would a low battery lead to absolutely no starter action at all or just slow cranking? I’m wondering if I bought a starter when I needed a new battery? Too late now since the new starter is in.
It depends on how dead the battery is. You will wonder about this for the rest of your life, and you will never know the answer.
Usually when a battery gets weak, you will get a multiple clicking when you turn to the crank position. There is enough ampacity to draw the starter solenoid in but when the contact is made to drive the starter motor, the voltage drops so low that the solenoid drops out. Then the cycle repeats causing the click, click, click, etc.
With a very weak battery, when you turn to the start position, the battery voltage drops so low that the solenoid will not even pull in so you get nothing. Usually any dash lights that would ordinarily be lit while you crank will dim to nothing. If you have the head lights ‘on’ they will dim out.
Hope this explains it.