These are not finite (tensed) forms but non-finite (untensed) forms heading subordinate non-finite clauses. Which specific form is required is determined by the verb heading the superordinate clause in which the subordinate clause is embedded.
I recommend you consider having it remodeled
Having here is not "present continuous"—that construction requires some form of BE before the -ing form. The -ing form having, without BE, is the non-finite form which the verb consider (when it is used in this sense) requires as head of its subordinate complement clause.
Traditional grammar, which characterizes the verbs heading non-finite clauses according to the syntactic role the clause plays, gives this use of the -ing form the name gerund—a non-finite form acting externally as a noun.
I'd rather not have it
Have here is not "present simple", but an infinitive, the non-finite form required by the modal verb would as head of its subordinate complement clause. The complement of any modal verb is always an infinitive unmarked with to.
I recommend putting it...
I recommend you to put it ...
I recommend you put it ...
Recommend licenses three sorts of complements:
marked infinitive (to put)
unmarked infinitive (put) — traditional grammar calls this a 'present subjunctive', but the form is identical with the infinitive in every verb. Since in present-day English this form complements only verbs which 'require' or 'urge' some action, it might plausibly be considered an imperative—which is also identical with the infinitive.