We have at least 2 gay Bishops and several transgenger vicars, one of whom has just been put forward to the Queen on the recommendation that she be made a Bishop (the vicar not the Queen). The Church of England doesn't offer marriage services for divorcees or same-sex couples but they do offer wedding blessings. The Church now offers weddings for transsexuals on the individual agreement of the local clergy member who would host the wedding, however, the candidate for marriage has to be marrying someone of the opposite sex and not be a divorcee.
B
No. Nor has the Queen, it's all been done in Her name by Her Bishops.
She was entitled to a church wedding however had she wanted to marry again she would have had to have a civil wedding with a blessing just as Charles had second time round because she, like him, was a divorcee.