Just so we are on the same page here. Your 350 had these same hot starting and hot idle issues. You have put in a newly rebuilt 400 and you continue to have the same hot start and hot idle issues, correct?

Regarding the hot starting issues, when the engine is warm/hot is barely turns over; but with a jump it has no problem and will start right up? I would start by getting the battery checked. Don't assume because it is relatively new that it is good. What are the specs on it - CCA? Your negative cable from the battery to the engine, I wouldn't be quick to change it, but I would take it off the engine and clean off any paint that might be there where it attaches. Clean any paint off and reconnect. Same where the ground pigtail at the battery is attached to the fender. Do you have a ground strap between the engine and the firewall? You should have, but with the engines going in & out it may have been left off. If it is there, make sure there is no paint under the connections.

For the hot idle issues, I think there are a lot of checks to do before replacing the carb. Is this the same carb that was on the 350? You said it was new, so if it is new and you still have the same issue I don't think it is the carb. When the idle slows and the car quits; is it sudden or does it gradually slow in idle and finally quits? Have you verified the choke is working correctly? What is your low idle rpm set at? Have you checked the fuel pressure? Is the fuel pump new? When the engine is warm/hot and idles down and quits - if you restart it will it idle at all or no? If it does, for how long?

Get that vacuum gauge out and let us know what the results are.
Cal