I don't get it.  

Alice wants to go on a date with Bob. Bob wants her to prove to him that she's vaccinated.

Alice gets offended that Bob doesn't trust her, and no longer wants to go on that date.  Or if she's actually invested, she offers to show the CDC card or health portal screenshot or some other low-effort indicator.   Most likely outcome: Bob is lonely, Alice finds other dates.

In no case will there be an equilibrium where significant numbers of people ask for or perform the craziness you describe.

It's a difficult process. This is important

Important also because it limits who can demand it.  It's almost certainly going to be measurably discriminatory against protected groups, which makes it a non-starter to ask for such a thing for any public or regulated commercial purpose.

Assuming both people have a normal dislike of bureaucracy, the equilibrium will likely be pure trust, or a selfie. Possibly of them actually getting the vaccine, or whatever paperwork they are given once vaccinated. Anyone so untrustworthy that you suspect they might have gone to significant effort to fake being vaccinated is not someone you want to socialize with anyway.

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1a. Because it's a difficult process, it can't be assumed that everyone who is vaccinated will be able to provide such proof. This makes it difficult/expensive to require proof from large numbers of people.

If Alice also want to date Carl, David, etc, does she have to do it again and again every time? Wouldn't it be easier to do it once and put in into a blockchain?

Ha, so we can implement immunity passports!

But how many people are actually interested in privacy preserving immunity passports?

I should emphasize that I'm not by any means suggesting that the above behavior should be normative. I'm merely saying that the governmental tools for one person to prove to another person or entity that they're vaccinated already exist.