What was a proposed alternative to Segwit?
The proposed alternative to Segwit was simply to raise the block size to be able to fit more transactions into every block.
What did Segwit solve more than just the scaling issue?
A bug in the Bitcoin protocol that caused transaction malleability. Which means that a transaction could be modified after someone had signed it and sent it away. This could be done through changing the digital signature in such a way that it didn’t change the validity of the transaction with all of its content still doing the exact same thing but the transaction ID got completely different, because the hash value that became the transaction ID had the digital signature as input to the hash function that made the exploit possible.
How is Segwit and the Lightning network connected?
Segwit solved bugs around transaction malleability that made other 2nd layer protocols like the lightning network possible when it became less risky with unconfirmed transactions. So simply put it made future second layer protocols possible on top of the Bitcoin protocol.
Are people, wallets and other services forced to use Segwit?
No, it was a soft fork which is making the consensus rules stricter. That means that every miner, full node, SPV or services that are not updating the protocol will still be able to validate the blocks that is created on the new soft fork chain.
My understanding is also that Segwit is backwards compatible so that one can still use “legacy” addresses to send transactions from and still be seen as valid tx’s. The soft fork and backwards compatible properties give the Segwit roll-out a smoother transition from the legacy protocol.