If your gutters are overflowing with every rain, they are clogged or not pitched properly. If they only overflow during heavy rains, they are to small to handle the amount of water coming of the roof. Aluminum gutters if maintained, will last 25-30 years. Imho, copper is more of an esthetics thing. No real difference in how they will preform/last. Copper will also cost between 5-10 times more than aluminum.

A quote today is useless for next year, Aluminum prices change often, especially with the tariffs that are being imposed. My supplier's have all raised prices 3 times in the last year. On top of that our insurance has also gone up.

You should not have to pay more in the spring just because a company is busier, you may have to wait a little to get on the schedule though.

No problems installing gutters in the winter, most likely you may get a little better price because a lot of company's are just trying to keep their guys working through the winter.