I know of the Genesis game but never played it (though I do own the newer titles in the series), but did some digging to see what I could find to answer your questions:

Am I imagining this games difficulty? I feel like I am making little progress and I’m always getting ganged up on. / Does this game require a lot of grinding or repeating tasks before moving on?

Everything I’m finding is saying yes to both, with the grinding resulting in the increased money, stats, contacts, equipment and practicing/refining your own strategy for completing runs. Luckily, there are multiple strategies for “quick” grinding, and certain equipment and stats that really help whether it be just straight up powerful stuff or specifically helps to deal with pesky foes like ghouls. Because there are different sections to the grinding with different strategies to doing it efficiently, I’ll leave the specifics out since they get a little spoiler-y in case you want to figure it out on your own. But, do let me know if you’d like me to reply with specific strategies that may be seen as too much of a walkthrough or too spoilery for some (they don’t seem like gigantic story spoilers or anything so if the grinding gets too tedious and you stop having fun, they might be worth knowing about).

Should I be killing these innocents I see on the street? I try to get shadow runs but they seemingly always involve killing ghouls, which bend me over and spank my samurai butt. I’ve put my morals on hold and have been tediously murdering the population for the little nuyen and items they have.

Luckily the Karma explanation section on the Shadowrun strategywiki actually explains this pretty well. I went through it and it looks like a straight up explanation without story spoilers so I feel comfortable just linking it here for you: https://strategywiki.org/wiki/Shadowrun_(Sega_Genesis)/Karma#Karma

-is the samurai class the all-rounder character or should I just restart as a shaman or netrunner?

I found a really good explanation on the Shadowrun wiki: https://shadowrun.fandom.com/wiki/Shadowrun_(Sega)/Archetype. Thankfully this also gives a much needed explanation of the differences between classes without spoilers. It seems like this info is very important to know upfront depending on what your playstyle is like.

As far as the game itself, I did not play the genesis version but definitely looked into it after I got the Shadowrun trilogy that GOG games gave out for free a few years back. From what I found, the Sega Genesis version is superior to the SNES version in a few ways but a lot of people enjoyed both. I really like the style and concepts presented in the series and the newer Shadowrun trilogy seems to have taken all of that and improved on it, with each game getting better and better at giving the experience intended. I’d say it’s worth taking a look at if you enjoy the Genesis game, or if you really like the elements of the Genesis game but aren’t enjoying it so much, the newer series might be what you’re looking for.

Edit: Spelling, grammar and formatting.

Edit 2: Also, here is a link to a PDF of the original game manual for the Sega Genesis version, which helps navigate the UI and buttons, etc., just in case you need it since manuals were kind of a big deal back in the day!