You need "of" in standard English.
Consider the following sentence:
The monster (that) I saw in the cave
was hideous.
That is a relative pronoun: it refers back to the monster (the antecedent), and it introduces a subordinate clause of which it is itself the object, that I saw in the cave (the relative clause). We could use what here:
What I saw in the cave was hideous.
As you see, what has swallowed its antecedent: the relative pronoun what always does that, because it always includes an antecedent. If the antecedent must be specified, what is impossible. Apart from that, it functions just like that or which. In your of sentence, if I were to replace your what by a that, it would look like this:
I can't confirm all of [the things]
that she said.
I was forced to supply the antecedent that was hidden within what, because that requires an antecedent. I made something up, these things. This all works fine. However, consider your sentence without of:
*I can't confirm all what he said.
If I replaced what, I'd get:
I can't confirm all [the things] that
he said.
This might look all right, but think about the antecedent: what is it here exactly? The problem is that the antecedent is not just the things, but also all: "what did he say? he said all the things; what can't I confirm? I can't confirm all the things". The adjective all is part of the antecedent, which is why it too should disappear if you use what:
I can't confirm what he said.
That is why your sentence without of doesn't work in standard English. There might be dialects in which this happens regularly, though. Normally it should be one of the following constructions:
I can't confirm all he said. (That is implicit.)
I can't confirm all of what he said.
I can't confirm what he said.
It strictly is neither intra- nor inter-group, as the context shows.
As you seem to know already, it is between two objects, and among more than two.
The differences referred to are between (each individual member of one group) and (each member of the other) -- a many-to-many relation, so we need among here.
intra: The juniors are quiet, but the seniors fight among themselves.
inter: There will be a friendly match between the seniors and the juniors.
Our case: There are friends among the juniors and the seniors.
inter, 1-to-1: There will be boxing contests between the juniors and the seniors.
Best Answer
Yes, take part in is proper. The right preposition to use with an idiom such as take part is often somewhat arbitrary and determined by convention rather than by strict adherence to the definition or "rules" of the preposition. Take part may be derived from or otherwise related to partake or participate. Both words take in as a preposition, which may have led to take part in.