I know Monero is dynamically scalable, but everyone knows that simply increasing blocksize does not make a blockchain scale.
Saying "everybody knows" doesn't really cut it. Who can say what kind of hardware will be available by the time usage of Monero network gets to the point where Bitcoin is today. It's really not possible to answer this honestly without backing it with proper research and hard data. The only thing we can say is that blockchains are horrible at scaling but they can scale. However, that doesn't mean we shouldn't consider 2nd layer solutions.
Maybe reading this post from a well known contributor hyc would provide some better insight.
Nodes will still need to download the entire blockchain to function. The more transactions happen, the faster this will grow.
Some data like range proofs can be pruned, but that would only help storage but not bandwidth. Storage is cheap and practically unlimited. Network bandwidth, however, is not so focus should be there.
Bitcoin is trying to solve this with Segregated Witness, to make each individual transaction smaller.
That's a misunderstanding, see here.
Since signature data (witness data) is stored outside the transaction (and outside the standard block), it means that that data doesn't have to be counted towards the block size.
The main thing SW fixes is TX malleability which Monero never had to begin with, and which is required to make 2nd layer work. Yes, it will increase TX throughput but that's because the network will accept some extra data along the 1MB block but the data still has to be transmitted and recorded. Trick is, that extra data is going around the block size cap, but the total amount of data per block cycle will now be bigger than 1MB.
I'd like to know how much bytes are needed now (on average) for each transaction, and what the roadmap is to decrease these transaction sizes (or go Lightning like Bitcoin).
For a normal TX it's about 13kB. That's about 25x normal Bitcoin TX IIRC. There are some optimizations to range proofs being worked on by Luigi, so the TX size could drop by 50% (preliminary estimate). At the same time, multi-signatures are being worked on, which is a pre-requirement to enable 2nd layer solutions if I understand correctly.
Another optimization is regarding block propagation, which is being tested at this time. It's similar to "compact blocks" in Bitcoin and should reduce the bandwidth footprint.
For long-term, there's the "Design and Development" road-map, which hasn't been updated in a while but can give you some idea. Another idea which has been mentioned was sharding the blockchain, but there's nothing of substance as of now.
Again, nothing is being worked on currently, but seeing how other projects are experimenting with solutions like lightning network, I believe this project will use some of those solutions as well in the future. Thing is, there's no urgency considering that level of adoption is nowhere close that of Bitcoin. I guess it can be the next research goal after multi-signatures are rolled out.