You should definitely still think about the session cache. Even though for the most part they have sticky session, the failures you mentioned in your second question can lead to lost sessions when they get shifted. It depends on how much you really use session and what a lost session will mean to your users experience wise.
If the underlying VM hosting the Web Site dies, the traffic will get routed to another VM and another IIS Process will be spun up to handle the request. If you have multiple instances set up already then it would get routed to another instance after it realizes the other one isn't there anymore. Note, based on my understanding, if you running in Standard mode you are running on a dedicated VM. You can control which of your sites within the same subscription and data center sites get moved onto this VM if you wish. This is how you can get a lot of cost savings if you run many sites.
Check out this Azure Fridays episode: http://channel9.msdn.com/Shows/Windows-Azure-Friday/How-Azure-Web-Sites-Scale-with-Stefan-Schackow