4. The bottom corner shows "not connected yet."
--snip--
here's your problem (in all likeliness)
If you click on the bottom corner, you *should* be able to pick a node manually (eventough the automatic selection *should* work). If this does not work odds are you have a network problem in your internal network setup (well, problem is a big word, your network can work as designed if it's designed not to allow connections to electrum nodes)...
You can always try tor if you cannot modify your internal network setup
https://electrum.readthedocs.io/en/latest/tor.html
As for your other question (how to verify electrum's signature), there's a cool guide in a pinned topic:
https://bitcointalksearch.org/topic/guide-how-to-safely-download-and-verify-electrum-guide-5240594
Your question is solved in the "Download and Verify Electrum" topics
Just to offer your some piece of mind... AFAIK, odds are very big the only problem is that your electrum client is not connected... As soon as you fix this issue, everything should work normally, if it's just a network problem, your funds are probably not at risk.