There is no name for them.
They are not obliged to do this, but if the Mother of the Bride is in reduced circumstances , and the Groom's parents can afford it, it would be a lovely Bride's Gift from her new in-laws.
It does not really matter whether it is the groom or bride's parents who call. In many cases the groom's parents call, but, etiquette does not dictate this.
sure,they should wish the best for their children together.
No it is not mandatory. Some parents (could be the bride to be's parents) may have the groom, bride, their parents and the wedding party out for dinner in the evening a few days before the wedding, but this is not mandatory either.
The Bride and The Grooms - 2009 was released on: USA: 24 April 2009 (limited)
Since it is the bride's mother who is sending out the invitations, which is customary then the return address should be to your parents or your mother.
Up to you
Usually the bride's parents will phone the groom's parents and the six of you would get together for either dinner at the bride's parent's home or, go out to a restaurant.
Most guests already know who the bride and grooms parents are and some weddings have a receiving line where guests congratulations the bride, groom and both sets of parents. If there is no receiving line then no, the bride and groom do not introduce their parents. The father's of either side may wish to make a toast to their new daughter-in-law or new son-in-law.
You would send the card to the bride's parents home. It would be addressed: Mr. & Mrs. John Doe (newly married couple) c/o Mr. & Mrs. Joe Smith (bride's parents name) (Bride's parent's address.)
At Hasidic and Orthodox Jewish weddings, the groom is escorted by both fathers and the bride is escorted by both mothers. At other Jewish weddings, the bride and groom are escorted by their parents with the father on the left and the mother on the right.
For the lovely wedding, if the Bride's parents pay for it, and for the gift of their daughter and themselves as a welcome addition to their family. Your children's wedding day is a time to be on your most welcoming behavior , even if you have the slightest reservations.