Are there any activities (in the Church or Temple) that a person who converts to the Church of Jesus Christ of Latter Day Saints cannot participate in that someone who is born in the church would be allowed to do?
We've never expected people outside our faith to live by our standards, and that would also apply to converts. As long as they've truly changed and are willing to live as Christ has asked us to live from then on, once they're baptized, they're on equal footing with the rest of us.
I've heard non-Mormons refer to "prominent Mormon families," but we really don't have any. And even if we did and knew who they were, it wouldn't matter. In our eyes, righteousness is determined only by individual choice, not by the choices of our parents or grandparents. A convert is just as likely to choose to obey God as someone born into the faith.
As for not allowing them to participate...new converts are asked to wait a year before they can participate in certain aspects of the temple. This is more about making sure they've had time to understand the doctrines relating to the temple, though, than it is about them not being worthy. We try to give new converts time to adjust and ask questions in the first months after they're baptized, because of course nobody can understand everything all at once. We want them to at least have the basics down before they enter the temple, because if they don't, going just won't do as much for them.
That's pretty much it. They probably won't be asked to be bishops in their first few years, but even that has happened in rare cases. They can even go to the temple in a limited capacity the second they're confirmed. There really is very little they can't do.
No, except for the 1 yr rule, and that also applies to lifetime members who have been inactive.
As for converts participating, look at Dieter F. Uchtdorf, he's a foreign born convert to the Church and yet there he is the second councilor to the President of the Church. .
There is nothing that you will be excluded from, so no, anything that an adult Mormon who was born into the church can do an adult Mormon convert can do to.
Not much... course maybe not going through all the programs the church has such as Nursery, Primary, Young men/Women and Seminary. But if a person who is a convert gets a calling in one of those then they also get to experience or participate in it. but other than that, those are programs that are basically age based.
My mom who is a convert sort of envies the fact that I went through all the church stuff as a kid growing up such as, Primary, Young Women's and Seminary. She thinks I know more about our doctrines and teachings because of it. I don't know, I think that's a matter of her just studying more of the scriptures and stuff.
Everyone in our church is a convert at some point. Those born into it must gain their own testimony and witness that this is the true church of Jesus Christ. As for the activities in the church; all members are welcome to participate and be part of the activities of the church. Attendance in the temple though is based on worthiness and so all members, converts included must obtain a temple recommend through their Bishop and Stake President (leaders in the church). New members can get a temple recommend after one year of their baptism.