Hmm how do I know if he is bringing or taking them since "tar med sig" could mean both?
You can't really, so we accept both options.
Do note, though, that it's "takes" in the "takes with him" rather than "grabs" sense.
Portuguese speakers confused by this may associate 'tar' with the english 'take', and 'tar med sig' with the portuguese 'trazer'.
You'll notice 'tar med sig' sounds a lot like 'trazer consigo' or 'trazer com ele próprio', which is ofcourse redundant in the portuguese language, but it allows for immediate assimilation.
You can then use this to help you assimilate other swedish verbs that use this type of construction.
Since you like using definite form instead of possessive, does this mean that they're his children?
Yes, the most likely interpretation here (and an accepted translation) is that they are his (own) children. It doesn't have to be, but that's what people will assume if you say this with no context.
Is there a non-explicit way to say/imply "he brings children that are not his"?
Yes, just like in English: "he brings children" instead of "he brings THE children".
"Han tar med sig barn" vs "Han tar med sig barnen"
Just to clarify, you'll know if it's one child if it has "ett," and many if it doesn't. Barnet would be singular as well, while Barnen would be plural
No. You have to use "Sig".
"Ta med honom" means "Bring him", referring to someone else.
Is there a difference in Swedish between 'he brings the children' and 'he takes the children'? I know that there is tension between British and American English with regards to bring/take, and would appreciate some clarification.
Yes, with "ta med (sig)" and "ta" respectively. In certain contexts, "ta" might work on its own, but it can sound very worrying to say that he takes the children if not used right.
what is the difference between 'tar med sig' and 'har med sig'? How interchangeable are they?
That's literally "take with (his/her/its)self" and "have with (his/her/its)self". The difference between going and being.
so sin barnen would be His children and sig barnen could be children in general?
sitt barn (single child) or sina barn (multiple children), but otherwise yes, that's correct. I will say though that even when it's just barnen I would presume they're his own. :)