1965 Valiant slant-6 225 carburator

No idle is a fairly generic carburetor problem; you might want to try this technique for openers and see where it gets you.

I can't think of a good reason why a choke would make the difference between the hood closing or not closing (though you've got me wondering what's present for an air cleaner!). And "the carburetor is modified" makes my beard stand on end. Fortunately there's a guy who specialises in 2100s, including modifying them for non-spec applications; see here. He can probably help you positively ID your carb and sort out whatever problems and improper-conversion issues you're having. If the carb's configured to have a built-in automatic choke (the kind with the round black bakelite housing and the heat tube) then probably that's why there's no active choke on it: noplace to run the heat tube on a Slant-6 unless you happen to have a particular exhaust manifold used on cars with the throttle anti-ice system (some Canada-spec cars '61-'69 and
all US-spec '69s), which you probably don't. Easy and much more satisfactory fix rather than rigging up a heat tube is to fetch an Electric choke kit for your particular carburetor once the linked specialist has helped you ID it.

You might please post some photos of your present setup, both with and without the air cleaner installed, from a few angles. There's probably stuff additional eyes will spot and be able to point out for you.

(BillGrissom, the '60-'66 A-bodies used the rotating-rod throttle setup except V8 cars (as you mention) and '65-'66 Slant-6 cars with factory A/C got cable-type throttle hookup.)