It's worth noting the picture dictionary is outdated in subtle ways. An SDG flag is granted for a raw goods count <10. You get more raw goods for every non-perfect/amazing judgement. Good = 1 Raw Good, Average = 1.8 Raw Goods, Miss = 2.4 Raw Goods, Boo = 0.2 Raw Goods. A perfect score (AAA) is 0 raw goods. So...
AAA > Booflag (1 boo, 0.2 raw goods) > Blackflag (1 good, 1 raw good) > omni/rainbowflag (which I'm not certain are actually shown in game and total to 5.4 raw goods) = SDG (<10 raw goods) > FC (0 misses and >10 raw goods)
FC* is given when you have previously full combo'd a song, but then get a better (more accurate) score on it that is not a full combo or another rating.
Average rank is just that, an average of the ranks you have gotten on every public (non-token) song in the game. Better scores will reduce this number as you will get higher ranks on songs than others, up to rank 1 which is generally a AAA (with only very high difficulty exceptions). Lower average is better.
Average rank rank is how your average rank compares to others. You moved your average rank lower than 1002 other people's.
The FFR wiki faq is also a really good thing to look through for more info on this stuff
https://ffr.fandom.com/wiki/FAQ