No, it's not permissible in proper English to omit the conjunction. Doing so creates ambiguity (is the rule that they have to be all of those the items in the list, or only one?). Sure, you might leave out the conjunction in casual speech or writing (it's totally fine to speak or write ungrammatically, accepting that context clears up the ambiguity created), but that's not relevant.
It is frequently permissible in proper English to omit the conjunction, just as it is permissible to add additional conjunctions. Asyndeton and polysyndeton are well-recognized rhetorical techniques that can be found in a variety of texts, from "I came, I saw, I conquered" to "Government of the people, by the people, for the people."
Statutes are no exception — plenty of rules say something like "plaintiffs may be joined if" and then go on to list various conditions, without including a conjunction before the final item in the list. The legislature probably should not do that, because (as you mentioned) it creates ambiguity, but it is not impermissible or unknown in formal writing.
Mixing gerunds and non-gerunds isn't a grammatical rule at all and doesn't indicate anything.
Yes it is — parallel structure is a well-recognized rule of English grammar. If the legislature is writing a list and chooses to break the pattern, the use of a different form should carry some meaning.
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