I don't blame you for not wanting to bother with PC emulators. In my own experience, at least, DOS box is totally stupid, why they had to or chose make it mimic how DOS use to actually work is beyond me, but it's completely not intuitive or easy to use. Compared with most console emulators which are extremely easy. Just open up, load game, play. Simple and easy.
As geoo explained, there are various means for DOSBox that can help make it easier to run games, at least for specific games. Beyond that, you really can't blame the emulator on this: DOS games in general (and more generally, DOS programs) were never as easy setup-wise as "just insert catridge and press Power button" console games, which being special-purposed devices were of course designed to make that part hassle-free. So blame DOS if you must, but the emulator is merely emulating the same pains you would've gone through back in the days when Windows doesn't exist yet.
In any case, it sounds like the issue now is not merely running the game, but that there are performance/compatibility issue running DOSBox together with whatever recording software is being used. There's really nothing to guarantee that a particular console emulator would fare any better in that regard, you just have to try and see if you get lucky or unlucky.
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As for version differences:
- the "original" versions in DOS/Amiga have 60 lemmings per tribe, rather than the 50 in Genesis and 40 in SNES, though I don't believe this has any meaningful impact to any levels.
- the SNES version definitely has a lot of changes, "dumb down" can be a fair assessment compare to the DOS/Amiga versions. I don't recall the Genesis version specifically but from skimming walkthroughs, I think it is at least closer to DOS/Amiga compare to SNES. If I have time I can try to find and call out any specific levels that have important differences. There is a known bug on Genesis where walls and ceiling almost but don't quite extend to the top boundary of the level, allowing you on some levels to bypass some parts by walking on top of the ceiling. But I believe this only really impacts like 2 or 3 levels, and you can simply restrain yourself from exploiting this bug in your own playing.
- I also vaguely recall that some of the console versions have better selection support (eg. directional select) compare to DOS/Amiga, though I no longer remember the exact details. It shouldn't matter too much though in most levels, but you never know when it may come in handy.
I'd imagine you definitely want to try playing it on a setup with mouse support if at all possible, since a good number of Lemmings 2 levels feature wind-driven and similar skills where you need to constantly adjust the cursor position to properly direct the lemming's movements. That will probably be easier to do with mouse.
