Yeah it works the same in every single neighbouring language, s is always the normal s(sz for us) and s+something is sh(s for us), i actually dont know where this convention comes from as the y marking for soft sounds is actually logical except for gy but thats a completely exceptable way to write it.

As for the szeged ö the sound is actually the exact same as other the other ös, but yes in the other dialects they are extra sounds that dont have their own way of writing. Also in standard hungarian this is not present so you wont hear it in budapest, northeast and from younger speakers in larger cities.

I usually pronounce bicikli with with a pretty solid k but i can easily imagine a lot of people dont, ive never actually payed attention to that specifically. The reason its bicikli but not lakzi is because of certain phonological effects having to do with “l” being much closer to a vowel than “z”. For bicikli/bicigli the voicing is just at a different point while for lakzi you would need to articulate a k, then start voicing, and then articulate a z and then transition to the i which is much harder to do. But dont worry there are thousands of examples like this that very much have stupid rules for spelling.