Misplaced modifiers, are another thing, though: often confusing.
So are misplaced commas...
There is no infinitive in those examples. Those are merely compound verbs formed with a main verb and an auxiliary. Split those all you want, according to "the rules." ("I will gladly support you."
But when we write "to + verb," now it's not a verb, now it is an infinitive, a verbal, functions as something else, bla, bla, bla, see post above. I am in full agreement that they should be deemed splittable.
No, that's the full
infinitive. The bare
infinitive is just "go" or "support" or "ask," the raw, uninflected form of the verb. Look it up -- I did.