If Peter was the first Pope, how do you explain the fact that he was married? (Mark 1:30)
Okay, Peter had a mother-in-law. I assume that the argument would be that since Peter had a mother-in-law, he was married; since he was married, he can't be the first (Catholic) Pope. Let's not dodge any punches here; how about we throw in 1 Corinthians 9:5 as well? Paul writes here about the right of the apostles - priests - to take a wife, like Cephas (Peter).
The foundation of the argument against Peter being the Pope, as far as these verses are concerned, is that priests cannot be married. If Peter was married, then he wasn't what we would consider a Catholic priest. If he wasn't a Catholic priest, then he certainly couldn't be a Catholic Pope.
It's at that foundation that the argument falls apart. I suspect many Catholics aren't aware of this - I wasn't for some time - but priestly celibacy is not dogma. It is a disciplinary decision that originates primarily in St. Paul (1 Cor 7:38).
Jesus does support the idea (Matt 19:11–12) but not to the exclusion of marriage - some people are called to celibacy and some to marriage. It's interesting that Jesus doesn't specify which group of people is called to which lifestyle; I'm assuming from that - and this is just that: an assumption - that the two groups are not simply priests and non-priests.
What we have, then, is a decision made by men which can change. From those two passages, in fact, it must have already changed at least once - from "priests can marry" (though it may have been more complicated than that, of course) to "priests remain celibate". The decision is based on the words of Jesus and the recommendation of St. Paul, so it will not be changed lightly; but it could be changed.
The issue of priestly celibacy aside, the "tough question" centers on Peter's marriage disproving his papacy. That argument, as you can see, doesn't hold. If being married and a priest is not against Church dogma - not against anything in Sacred Scripture or Tradition - then it doesn't preclude Peter being married and being the first Pope.