Wacky electrics on Honda Accord '05

Recently my car has been having starting problems and other weird electrical glitches. I’ll begin with I use my turn indicator switch both arrows light up with the direction I want to go in being brighter and clicking faster. Second problem is I step on the brakes and both arrows light up. At one point two days ago I tested the hazards button and and lights on the ac panel started flashing but it has not happened since. AAA battery people were stumped. Car is at the mechanics now but any help would be much appreciated.

Sounds like a grounding problem. You need a good mechanic who can work on electronics.

For the brake to turn the turn signals on, there’s got to be a short in one of the rear turn signal/brake light bulbs - the short would need to be between the turn signal and brake light contacts. With that short, it will also flash both left and right turn signal lights, if you signal a turn into the shorted side.
Take both bulbs out, one at a time and see which makes the problem go away.
The flashing at a different rate can be explained by the flasher circuit being what’s called a ‘relaxation oscillator’: Its rate is dependent on the electrical load of the flasher relay. Since you have more bulbs on the circuit, the load has changed and it flashes at a different rate.

Here’s the schematic, btw.
Notice the red mark. That could be where your short is, there or in the other bulb/connector/wiring. It would kinda do what you describe except that it wouldn’t turn the dash signal lights on in both directions - only in one direction, until you brake (that’s when it will likely stop flashing and just turn on). It would turn both those dash turn lights on when you brake.

If the car has a wire harness for a trailer check the connector to see if there is some corrosion on the connector pins. Somehow the circuits are getting crossed over and that would be the most logical point if you do have a harness for a trailer. If you don’t have a trailer harness check the rear lights for some water damage to the sockets or to the wires going to the rear lights.