If the reviews on newegg for this this card can be trusted, then a 350 watt supply should be sufficient. A lot of people on the reviews are claiming to run it with sub-300w PSUs. From what I've been reading, the 7700s are bar none the lowest power-users in the budget market right now, so I'd say this is a good choice, but I could be mistaken there. What's the model number on your Dell PSU? And how old is it?
You're good. Dell may use undersized PSUs, but they aren't junk (generally Delta or FSP), so you can expect that one to actually be capable of 350W. The HD7750 only uses 56W running flat out (i.e. benchmarking; games will use less), so you'll be fine.
Yes, my sister bought this card for herself, and I am amused with the results, can be used to watch HD movies and also for gaming (skyrim, Diablo III, BF 3) with no problems!
The 350W psu will be fine.