Can't find the answer to this online, and just noticed this when obsessively trying to find anything that could turn around the mess I made.

Player is on the 40 man roster, but not on the big league roster, and has options.
Player is traded to another team.
That other team assigns that player to the minors.

Should that player be on the new team's 40 man roster (if not assigned there by the new team)?

I noticed that the player I acquired is not on my 40 man (which is to my benefit, really), but was wondering if it was correct.

I wouldn't think so.  If he has options, you shouldn't be forced to "inherit" his 40-man status from the other team.

Ok.
Question 2, if that's true...clock is started when he's added to the 40 man. Is it still running now that he's not?

Yes, I believe so.

No, you have to assign him to the 40.

Not sure what clock you're asking about but an option was burnt when he was assigned to the 40 for this season.   That will not change.  If you don't add him to the 40, he will either be a FA or eligible for the R5 after rollover assuming he's not a 2nd year player.

Ok, I'm in line with Mike's logic, but...
http://www.whatifsports.com/HBD/Pages/Popups/PlayerRatings.aspx?pid=3120928

Apologies for not being able to hotlink

Assigned to 40.
Traded
Currently sitting in AAA with 3 options, not on 40.

Next season, it's definitely going to make me add him to the 40 or lose him in Rule 5. But, will his option years be down to 2?

He will have 2 options.

IOW, if you add a player to your 40 for one minute, he burns an option if he's in the minors. 

Ok.

So, if there is a bug at all, it would be that I, theoretically, could have 41 on my 40 man roster if he should have been on my 40 man right now.

Options will move correctly, the need to add him or lose him next year will function correctly. Correct?

You won't be able to add him to your 40 if you already have 40.   The only time you can exceed for is with removal from the 60 day DL.

If he's a 5 year player, he'll be a FA if you don't add him to your 40.  Otherwise, you can wait til next season but this year's option is burnt.

Let me narrow down the piece of which I'm still not clear;

Should he by rule be on my 40 man roster because I acquired him with him being on a 40 man roster?

The rest I'm good with now.

No, he should not.   No more so than a BL player with options should be forced to be added to your 25 man after a trade. 

I'm good now!
Thanks.