Well, there are huge differences between point-and-shoot models but your budgets pretty much limits you to the nearly indistinguishable bunch of lower-quality ones.
There is not much to do about losing a camera but for other obstacles you can get a rugged model. Most are drop-proof from 1 to 2 meters (about 6'), waterproof from 3-10m (10-30') and completely dust and moisture proof. There are a number of such models, which I believe the Fuji XP10 is the cheapest and falls within your budget. Here is the list of water-proof ultra-compacts.
To get better image quality you will have to spend more. At least in $180 for the Fuji Finepix F80 EXR. Fuji EXR models have excellent image quality and generally faster than most point and shoot. Better P&S will cost you much more for the technologies they use (larger sensors, CMOS, brighter lenses, etc).