rampant vs rampantly:
You're all correct that "rampant" is an adjective (modifies a noun), and "rampantly" is an adverb (modifies a verb). But you could actually use either an adjective or an adverb in this case.
In this case, I think "rampant" or "rampantly" could both be perfectly fine. I'll strip down the sentence a little bit for clarity:
Kudzu has grown rampant in the southern United States. --> "rampant" is an adjective, and we're just saying that the kudzu itself is rampant
Kudzu has grown rampantly in the southern United States. --> "rampantly" is an adverb, and we're just saying that the kudzu has grown in a rampant (or uncontrolled or unchecked) way

for doing :
Two key things to remember:
(1) "for" is a preposition and therefore cannot have a true verb in its prepositional phrase and
(2) any modifier (including prepositional phrases) must refer to the right item.

A pen is for writing ('for writing' tells us what a pen is used for)
I picked up the pen for writing a letter ('for writing a letter' is fine as a modifier, but it incorrectly modifies the word pen- I have five pens on my desk, but I picked up the one that I use for writing letters)

here's another attempt at explanation:

* if you use "for VERBing" to explain the purpose of an action, that purpose should be (at least somewhat) indirect.

e.g.,
(nb: a "walking foot" is a component of a sewing machine)
i bought a new walking foot for working with leather.
--> indirect purpose. i.e., the purchase itself is not directly involved with leatherworking, although the two are indirectly (though rather closely) related.
i used the new walking foot to work with leather.
--> direct purpose.

the other legitimate uses of "for VERBing" that i've seen have mostly been idiomatically connected to particular meanings. moreover, essentially all of these constructions would work the same way with nouns substituted for the "VERBing" parts.
e.g.,
Clara was paid $600 for working 20 hours of overtime.
--> this one depends on idiomatic usage: you are paid for x.
note that this construction works identically if x is a noun: clara was paid $600 for her work.

https://www.manhattanprep.com/gmat/forums/when-is-use-of-for-objectionable-why-is-it-not-here-t16521.html

If we look very literally at (C), it's saying that the vine has grown rampant in the time since it was introduced in the 1920s. That makes perfect sense, and the word "since" is a reference to time.

But as soon as we put the comma before "since" in (E), the word "since" takes on a different meaning. The phrase "since it was introduced in the 1920s to thwart soil erosion" becomes a non-essential modifier -- and it seems to be giving us an explanation. "Since" now functions as a synonym for "because": "the vine growing rampantly in the southern US, because it was introduced to thwart erosion, has overrun..."

That's not quite what we're trying to say here: the vine hasn't grown rampant because it was introduced in the 1920s to thwart erosion -- it's just become rampant in the time since the 1920s.

Nasty and subtle, no? :(