"I work a little bit."
I was in Japanese in high school for nearly four years and my teacher taught us that すこし was used for a little bit and ちょっと was used if you were being less polite or telling someone off - ex. if someone asked you if you were busy on the weekend you'd say ちょっと。。。and it would be like a polite refusal as opposed to saying straight up いいえ which is considered very rude. My teacher also taught us that when someone is wrong, you can use ちょっとwhen you say ちがいます and that also makes it less forceful
It's actually more a difference of formal and informal.
少し is more formal than ちょっと.
So you would use ちょっと with your friends and peers, but not with your teacher or boss.
You would use 少し when speaking to strangers or people who you would use honorifics with, ie. Teachers, higher ups at work, police....etc.
these are called suru verbs in English resources, they are basically nouns with strong verbal properties to which you can just attach する at the end. So「仕事する／仕事します」is kinda like a full verb. Sometimes they leave the を particle in there for emphasis porpuses, I haven't found the logic in that, they say is about casual speech but they don't do it in formal settings either, so I have no idea about that. What I do know is that not every を sounds natural, sometimes it's better to leave these out in short sentences like this one.
For usages, I cannot really list you all, but most of the time (as I said before) they prefer 働く when speaking about the work/labor being done, while 仕事する is more to describe the job in a company, the position, the profession. 働く is a bit more friendly too.
Probably disjointed is a bad expression,「ちょっと仕事『を』する」seems grammatically correct but is not how they really use that を imo. I feel like if they are gonna use it like that they would probably say 仕事をちょっとする、but this is not natural for this expression and I'm probably just extrapolating from other cases.
Maybe you can translate them like this:
「ちょっと『働きます』」"I work a little bit" or "I labor for a bit"
「ちょっと『仕事する』」"I do my job for a bit" or "I carry out tasks for a bit"
「ちょっと仕事『を』する」"I do the task for a bit" or "I carry out the job for a little while"