What was a proposed alternative to Segwit?
Increasing the block size to 2mB.
What did Segwit solve more than just the scaling issue?
The tx malleability issue.
How is Segwit and the Lightning network connected?
Off chain usage is now less risky as the transaction can be validated at all times due to the fact that the signature can accompanied by the tx can not be changed anymore and will not change the tx id.
Are people, wallets and other services forced to use Segwit?
No, because it is still possible to store the signature inside the transaction, but the transaction will just have more weight and therefore need a higher tx fee.