That depends - I could easily imagine the government removing social distancing in pubs which had a certificate system, but enforcing it in pubs which didn't. I doubt they will but you can imagine the government trying to create those sorts of incentives.
Possibly, but I cannot imagine that the hospitality industry would voluntarily comply with a measure that would mean they have to turn large numbers of people away.
I would imagine that people like Tim Martin of Wetherspoons would be very vocal in his opposition to such a plan. Whatever you might think of Tim Martin, as soon as one or more of the major pub chains announces that they are not going to implement a vaccine passport system, many independent pubs are likely to follow suit.
In any case, social distancing is supposedly going to be abolished after June 21st, so removing social distancing in pubs which had a certificate system but enforcing it in pubs which didn't, would only be a short term measure.
This is before you consider the possibility of a court case from someone who cannot have the vaccine on medical grounds connected with a disability, who can then sue under the Equality Act 2010.