Unfortunately some babies are catnappers. It sucks. Sometiems they just grow out of it, sometimes they don't.
Soem babies also dont' feed well (if at all) if they're very tired. You need to try just as she's waking up (even if it's only a 20 min nap).
The on and off fussing could be any number of things - any illness around? Just a fussy stage. Who knows? Watch the wet nappies, that will tell you whether she's getting enough milk. The poo will come in time and it might be a big one
Do you use a sling or carrier at all? It doesn't really work for my DD, but my DS got some decent sleep at this age in the hug a bub (wouldn't sleepmroe than 15 minutes most days otherwise)