The carb screws coming loose is actually not uncommon, especially on older Carter designs. I had (and probably will have) the same problem with the BBS 1-barrel on my '68 Valiant. I just tightened 'em and called it a day for now since that carb needs a rebuild anyhow. A drop of weak threadlocker (blue) on the threads should solve it.
As far as dumping gas at the bottom, I assume that was coming out around the throttle shafts since there's no gasket or seam in the carb body that low. That could really only be caused by a stuck or sunken float, dirt in one or both of the needle/seat assemblies, or way too much fuel pump. Since this is the first time we've seen that to my knowledge, I doubt it's the fuel pump. I also kind of doubt a sunken float since it just appeared, but I can't rule that one out completely. A stuck float is more likely, especially with Edelbrock carbs being prone to rust in the float bowls due to substandard steel/plating on the float baffles.
However, what really bothers me is the "pouring gas out" part of your post. There's nowhere for gas to pour out below float level on an AFB. If it leaks around the throttle shaft(s) it should be seeping at most. Are the throttle shaft bores that worn? How much wobble have you got in the primary throttle shaft? Secondary? Usually the primary is the issue, and most often that's caused by too heavy a throttle-return spring. Over time, the back-and-forth rotation of the throttle shaft wears its bore into an oval or egg shape. That would've been an existing condition on your carb when you got it, since you've never even driven the bloody thing. Were it a "numbers" carb, it would be worth the expense of having the shaft bores bushed. An aftermarket carb without a replaceable throttle plate most often doesn't warrant that kind of expense.
This is why I almost never install a used carb without a full, fresh rebuild. I've done it in the past just to move the car, but otherwise they get a full rebuild right out of the gate. Carter AFB ("Edelbrock Performer") and Holley carbs are so easy to rebuild there's really no reason it shouldn't happen.
If you're looking at a new carb, the AVS2 isn't a bad choice. It's essentially what you've already got with an adjustable secondary air door, rather than a proper AVS clone. It uses the same tuning parts as the carb you've got, not that I think that's a concern. The secondary air door
might want some tuning, though, so that's another consideration. More potential tweaking, although in this case it's very simple.
When you re-sealed the intake, did the bolts go in very easily or were there hole mismatches/bolts that wanted to go in at an angle? Just glancing through, it looks like you forgot the tip about Permatex High-Tack spray gasket sealant on the intake gaskets, although that shouldn't be as critical with the end seals removed. The reason I asked about the bolts is because if they don't seem to want to align, that's usually an indicator of machining either on the manifold itself or the heads. That could cause a port mismatch or the surfaces to not sit flat against one another. Thicker gaskets are available for exactly that reason. Regardless, there's still a vacuum leak
somewhere. I managed 10" of vacuum with the Valiant at one point, which has a far more aggressive camshaft. I'd think you should be at 14" or more even with your cam being unknown. If the idle mixture screws have no effect, there's definitely a vacuum leak. Finding it is the problem. Check those throttle shafts for play first and foremost. Another thing to remember is that every port on the carb should either be connected to something or capped. Period. The sole exception to this would be the bowl vent, which is something the Edelbrock 1406 should not have.
I noticed this picture, with the mystery vacuum line to the choke thermostat:
I have no idea what that hose was supposed to accomplish, but I've never seen that on an Edelbrock. As such, I've no idea where they go inside the carburetor. Make sure the fittings at both ends of that hose are capped or plugged. The bottom fitting may not go anywhere important, which you can check with the engine running--put a finger on it, can you feel vacuum?--but if you're unable to do that, you could tap that hole and install a set screw or you could fill it with a product called Seal-All, which is the only sealant that will withstand fuel. Side note: Don't ever, ever use Seal-All as a gasket sealant. It is impossible to disassemble later. You were warned.
Disconnect every single vacuum line on the carburetor: the one to the distributor, the HVAC system, the power brakes, the PCV valve, and anything else. Cap all those fittings
at the carburetor with proper vacuum caps. If you don't have them, run to the parts store--they sell an assortment. Don't plug the hoses or use electrical tape or any other "well, I have this instead" half-assed solution. Make sure the vacuum caps aren't torn or rotten if you've got 'em lying around. Next, make sure the carburetor mounting hardware is tight. Yes, those nuts and bolts
do come loose from the intake now and then.
With all the vacuum lines gone, their fittings capped, and the carburetor verified to be snug to the intake, start the engine. Allow the engine to reach operating temperature; in other words the temp-gauge needle has stopped moving entirely. I've mentioned this several times and it's worth mentioning again: You can't accomplish
anything with the carburetor until the engine is fully warm (and the choke is 100% open). Once it's hot, try again with the idle mixture screws.
If the mixture screws now work, then something you disconnected from vacuum is the problem. As you reconnect each vacuum hose, try the mixture screws. Did they stop working or seem less effective? Then that hose, or the component to which it connects, is the problem. It's important to reconnect one hose, try the screws, and check your results. For instance, reconnect the HVAC hose. Try the screws. Are they just as effective? If they're not, then your issue is in that vacuum circuit. If they still work, try the vacuum advance and check the screws again. Still workin'? Move on to the brake booster hose and repeat. When the screws stop doin' their thing, there's your problem.
The two major suspects here would be the brake booster and the PCV valve, since either is a massive vacuum leak if not sealed. Yes, boosters can fail internally and still seem to work OK but allow huge amounts of air into the engine. The PCV valve is an "engineered" vacuum leak, but the right valve is important. I wouldn't expect the HVAC line to cause that much issue, but then again the orifices in the idle-mixture passages are tiny.
If the mixture screws still have no effect then something's cracked, warped, or not sealed on the engine itself. Check the intake and carb closely for cracks. Make sure the throttle shafts are not wobbly. The idle screw should be turned just enough for the engine to idle and no more. Double-check that the
secondary throttle bores are completely closed. Replace the carb gasket(s) if it gives you peace of mind; adapters introduce twice the potential for a leak. Jerry's "proper gasket for that manifold" is approximately right, being the correct shape and in
some cases thickness (not your case), but that partictular one should go right in the trash. Never reuse gaskets. The best gaskets for your application are either those provided with the adapter new, or the non-stick blue gaskets that are about 1/16" thick. The lower one (between the adapter and intake) should be shaped like Jerry's thick one, although it may have four holes rather than an open center. The other gasket in that photo is the right shape as well, and closer in thickness to what you want.
You said you've got a '71 intake, which doesn't have any "native" or built-in potential vacuum leaks like the '74-and-later manifolds with EGR. That leaves us nothing to chase on that end, but it simplifies things... one less thing to go wrong. There's only one vacuum port in my '71 intake, on the driver's side behind the carburetor flange. On mine there's a 1/4"NPT pipe plug in it, placed there by Mother herself.