What about solution number 3., Bitcoin is changed into a system that can handle more transactions/handle them faster. Would that be doable?
When do you think we would hit the limits for, say, a $20 transaction (BTC still being useful for it)?
We can change the block parameters in Bitcoin to support a higher transaction volume, but that only works up to a certain point. For example, if you allow 1000x more transactions in the blockchain, then the blockchain becomes 1000x bigger (20 GB -> 20 TB) and it's too big to host on one machine. Then you need some centralized service to host the blockchain and you lose the decentralized nature of Bitcoin.
You can do some optimizations to make the blockchain smaller (e.g. pruning old spent transactions), but that still doesn't get us the whole way there. I can see Bitcoin supporting, say, 500 transactions/second with some modifications, but I don't see any way to get up to 5000.
Off-blockchain transactions are another way to solve this, but then you are back to a centralized service to process those transactions. Inputs.io was supposed to do this but they got hacked and had most of their coins stolen.
Bitcoin can support $20 transactions as long as the transaction fee stays less than ~$1 (I would consider a 5% fee to be an upper limit for a practical transaction). We could get to that point if the value of 1 BTC got into the $5000 range, or if the transaction volume got high enough that miners started requiring higher transaction fees (e.g. 3-4x the current transaction volume).