Real quick - The flexible pipe is used where you have to to around rocks, bolders, trees, bushes, and the like, and you CAN'T go in a straight line. The solid pipe is what you use in softer soil where you have a straight run. ALSO, it's used where you may run 'heavy equipment' over top. (To be safe, in BOTH cases, try and run the pipe AWAY from where a car might drive, even by accident. If you crush/pinch the pipe, you're in deep trouble.
Yes, after you dig the trench, the FIRST thing you do is lay in filter fabric. You need enough to fully 'wrap' the gravel and the pipe. That's all. It's OK if there's excess that needs to over-lap. Beyond that, just cut it off. Check out this link on How To COnstruct A French Drain. Read this link then click to the link about installing the drain.
Yes, you can connect a gutter drain to the black system. I suggest you get a back issue of The Family Handyman Magazine and ask for the back issue that talks about installing a drainage system for your gutters. It's very good and meant for the DIYer.
I'm glad you're doing the research first. Get the back-issue. It's worth it! My best to ya and hope this helps.