I'm not expert or fans of LN so i can't answer all question
2. It looks like the most efficient evolution is a network where everyone connects with only a few, big, stable nodes?
On LN? Yes & it already happen naturally as merchant/seller usually become "big" nodes.
3. What happens if everyone wants to get their bitcoin out of the network at the same time?
If you meant LN network, then main-chain will be flooded with transaction about closing LN channel
4. What happens if the big nodes decide to just go offline??
There's no thing such as big or small nodes. If some nodes are offline, bitcoin network will run just fine.
But if you're talking about LN nodes, then user funds are locked for some times and there are 2 possibility with transaction routing :
1. User simply route their transaction through another nodes/channel
2. User is forced to make new channel
6. If the Lightning Network ends up being what everyone uses, who will use the main network? Also, does that mean that mining might be impacted?
You still need main network even if everyone use LN. You still need to create on-chain transaction to open and close LN channel.
7. If block rewards keep reducing but transaction volumes don't increase, will that impact mining? Or is the idea that fees will high enough to pay miners? Does this also assume that the value of Bitcoin must be high?
The scenario you mentioned is possible, but if both don't happen then mining difficulty will drop when block rewards decreased