Skype only autologs out if you actually close the Skype app. If you want to do other things and keep Skype ready to receive calls in the background, just minimise it instead of closing it completely (you can minimise it using the icon in the top right corner next to the X).

You can only access Skype through the Skype icon, you can't use Contacts. The reason for this is that Skype stores your Skype contacts online rather than on the tablet, so you will see the same list of contacts whatever device you use to access Skype.

Regarding voicemail, do you mean voicemail from your mobile phone provider, or do you mean some other voicemail system?