Do you think a 1.77us gallon tank is good for a betta

It is a little on the small side in my opinion, but should suffice (a lot of people use 1 gallon tanks).

Bettas are tropical though and need higher temperatures. One of the problems with smaller tanks such as yours is that they can be difficult to heat, but I believe there are some mini-heaters on the market so it should be OK if you can get one of those.

I live in a cold place but in winter where I want to put my tank has a heater next to it and in the summer hot air comes in through the window and it is really hot and it is about 80 degrees farengheit

The heat has to constant 24 hours a day and most people's homes change temperature quite a lot throughout the day. You should really get a mini heater for the tank.

I have two betta's in 1 gallon bowls. Just be sure to test the tank for ammonia the day it shows to a 100% water change and then change it next time one day ahead. Mine gets a 100% water change ever 5 days. I have moss floating in mine.
Don't get to close to the heater don't want to be to warm. I keep mine in the winter time in the warmest room in the house. My livingroom.

do you know how much the heaters are since my mom and dad make the money I have to get something cheap.

Don't know about prices or brands where you are sorry, as I live in the UK.

At the moment you have the tank but no fish right? Get a thermometer for the tank, fill it with water and place it where you plan on keeping the tank. Check the temp every hour or two (probably best to do it on a weekend when you have a free day) and make a note of the temperature each time. That way you can tell how warm the water is and how much the temperature changes throughout the day.

Do you know if the heating is on throughout the night as well?

Thanks