Would it be a good idea to recall that Brainwallet on a hot PC?
That depends on how concerned you are about security. Once you type in the necessary details from your "brainwallet" on a "hot PC", you are immediately vulnerable to various malware including keyloggers.
So, would it be okay to use on a cold PC, if that PC is never connected to the network?
That depends on how concerned you are about security. Once you transmit a transaction, the public key of your "brainwallet" becomes public. You also have to be very careful when constructing your transaction and make sure that any change from the transaction is sent back to the "brainwallet" address.
And then, if the brainwallet is amazing and the PC being used to recall the keys is disconnected, then would I be safe?
That depends on what you mean by "safe". The private key would no longer be protected by RIPEMD-160 or SHA-256. You would reduce the levels of protection to only ECDSA. That isn't really a concern at the moment, but if there are new developments in the future that result in ECDSA becoming insecure, how confident are you that you'll hear about it and move your bitcoins before the exploit is used to take them from you? Furthermore, you'll be giving up some anonymity by continuously re-using the same address all the time as well. Is anonymity important to you?
Could I use that same brainwallet to receive bitcoins?
You could. It it up to you to decide how concerned you are with any loss of security and anonymity.
Could I safely withdraw funds from that brainwallet without compromising the original brainwallet?
Not sure what you mean by "compromising", but as long as ECDSA remains secure and you use a computer that has never been and will never be connected to the internet to create your transactions, you should be ok.
I ask that last question because I've seen so many tutorials where people go through such an elaborate process with Armory, Electrum, saved Javascript websites, etc. on a cold PC that never connects, and as soon as they use that highly safe private key on a connected PC to do some business, that private key is immediately invalidated as having been compromised or exposed to the wild.
Correct. That would be because they want to maintain additional anonymity and want the full protection of ECDSA, SHA-256, and RIPEMD-160. Some of them are also concerned about the possibility of accidentally failing to send the full balance of the "change" back to the original paper wallet address.
If I have to do that each time, what good is a brainwallet at all?
Brainwallets are generally a pretty bad idea, but assuming for the moment that you manage to memorize something generated randomly with at least 160 bits of entropy, it would be good for long term storage of bitcoins that you don't expect to use for many months or years.
What good is memorizing the private keys even?
Do people do that? Memorize private keys? If they do, I suppose it would protect them from theft of their long term storage.
Is there a way to withdraw funds from a private key without exposing it to the wild??
Yes, but it results in exposing the public key and eliminating the protection of SHA-256 and RIPEMD-160. It also results in a loss of anonymity.