Hi Lisa
OK - it's really common for babies you DD's age to have short naps like she is having. The wonderful ladies on the naps board can give you suggestions about lengthening naps - although often it is developmental and she will outgrow short naps around the 6 month mark.
Personally if I were having those difficult short naps but able to get a good decent nap out of my DC by having a lay down with them - I would do it.
I think your night wakings are probably more related to the fact that you are feeding down to drowsy before putting DD into her cot first thing at night time. I think she is developing that association of feeding - if not to sleep then almost to sleep - and that is disturbing her when she goes through transitions in the night.
I would suggest you try putting DD down a little less drowsy - and use pat/shh to help her fall asleep (yes, it does take a while - I think it generally takes around 20 mins for LO's to fall asleep). You could try adding a book or a short cuddle with Dad after her feed, before she gets into bed.
It won't fix problems overnight, these things do take time but they will get sorted out once your DD gets better at falling asleep completely independently.
Good luck
Jo