It's free. No catch.
They're compressed but not on a level anyone will notice. If you upload a file that's 4000x3000px it'll remain that.
I'm not aware of any easy way to download your whole collection again, but a third party script may be available.
Yes, you can keep your photos private. It's quite easy.
Yes and no. There used to be ads but they're either gone or not online yet for the refreshed design. Flickr used to make money from "pro accounts" which were 25$ a yr accounts that gave more options. When Yahoo bought flickr through, they basically stopped adding options that people wanted. The features request threads on their forums were endless and mostly unfulfilled.
It was quite sad. Now, they nixed the pro accounts and have a 1tb for everyone and 2tb if you pay for it, a lot.
So their income model, I don't know, but it's yahoo. Like this refresh will capitalize on a current resurgence of interest in Flickr but I feel they're refreshing and just going to leave it forever again.