1. Money for FA = money available now, this year
Money for extensions = money available for next year
They slightly change in meaning depending on the time of year. In the off-season, when FA starts, then they line up well. During the season, money for FA gets kind of adjusted, since if you have 10M available but are halfway through the season, you can potentially add a player who makes 20M to your roster (since you'll only add 10M to your salary commitments). And best not to try to understand things in that time between the off-season starting and when FA file
2. Budget will be more or less based on how much revenue you pull in. So if you draw more revenue, your budget is likely to rise. Pull in less, it might fall.
3a. The budget is for the year, so when the league rolls over to the off-season, that's when the new budget comes into play.
b. Mid-season, the main new budget items would be draft bonuses and international FA bonuses. You can obviously also use budget room to bring in more expensive players in trade, too.
So basically, once the draft and international amateurs are done, there's not really any other use for budget space, so if you can, you may as well use it. However do note that in a limited fashion, depending on your owner and settings, some cash can be carried over to next year, so you might not want to use it. But certainly if you have room, it might be worth it to find an overpaid player and improve your talent that way.