Oh S. - do you have my child? :) My little girl was the exact same way! Exact! Her crib is upstairs in her room... I finally started putting her down for naps in her Pack N Play in the dining room. That seemed to do the trick. I think she felt like I was close by. Plus, for us, it reinforced that naptime is different than bedtime. She got used to naps being temporary and nighttime being an all-night-through thing.
I think at this age rocking her to sleep is fine. Most peds say that babies don't get "trained" by rocking until 4-6mos old. As far as her waking up once you put her down, have you tried making sure she's in a deep sleep (the phase after active sleep) before you put her down? That usually takes about 20 mins after they appear "asleep". They are much less likely to wake once in deep sleep if you put them down. As silly as it sounds a good way to tell is to lift their little arm and let it flop. If they move or keep their arm somewhat stiff they are still in active sleep but if it flops right back down they're in deep sleep most likely. Maybe you're doing that already and just have a light sleeper... its just something that worked for us.
Well if none of this helps at all just know that someone else out there went through the exact same exhausting thing and that there is an end in sight - it will get better! Best of luck to you both!!