Hey Prince,
I think for the timing issue doing it the morning before travelling is a bad idea - you guys will inevitably need to be rushing around preparing for the travel, so you wouldn't get any time to "enjoy" the proposal.
I think doing it once you have arrived is a nice idea.
When you guys have done the check in, got to the room and ditched your stuff, you could ask if she wants to take a shower before getting ready to go for dinner. Once she's in the bathroom you order up some champagne from room service and ask her when she gets out the bathroom in her robe? It's a bit low key, so depends on your girlfriend. That way you can pre order more champagne at the restaurant for when you eat out to celebrate :D
I wouldn't advise asking in a restaurant - there are loads of women who would be really embarrassed at the public attention, so it's better not to risk it.
I also really like the idea of doing it on a post dinner walk along the beach. That would give you privacy, a nice memory, etc. Just make sure it's somewhere light enough that she can see you getting down on one knee!!
The only problem with this option is that it puts a lot of stress on you trying to find "the right" spot to propose in. One of my friends got dragged half way across London when her boyfriend proposed because he couldn't find "the right" spot. She was about five minutes away from getting a taxi home on her own when he finally asked!
Finally on the ring question, I'm afraid no-one here can answer that - it completely depends on your girlfriend. Hopefully if you are considering proposing, you know her well enough to judge whether she would prefer you to pick the ring, or to have a say in choosing one.
For example, my fiancé chose my ring on his own, and I was really touched at the effort and thought he put into it - and he chose exactly what I would have chosen, so it proves how well he knows me. Of course, I would have accepted the proposal even if he proposed with an onion ring, but it did mean a lot to me that he chose the ring. Other women would have hated not having any input.]
It depends utterly on her.