In particular I'd like to use XMR-Stak miner and use it maybe to mine Monero.
For Monero, forget about this, unless you have a huge huge huge mining farm. It doesn't pay off to wait to find a new block for too long, as opposed to sharing the reward together with other miners from a mining pool. Eventually, in the limit of very long time, both rewards should be similar, but in the case of solo mining it may take an eternity to see some money.
But if you want to mine some new not yet known coins, it may be wise to do it solo.
Is it possible to do this? I understand you can configure a wallet to connect to a remote node. I wonder if its possible to do a similar thing with a mining node.
In any case you have to download the entire blockchain and be very very good connected with nearby nodes so that when you find a node, your result is quickly propagated.
Maybe I didn't make it clear at the start, I would only mine using a mining pool, not solo as I know that is not worth it.
So if I mine using a mining pool (ie with my own server, not using a paid-for remote mining service where its all done on the server run by the provider) then I don't need to download the blockchain?