Problems, yes. The same problems? ...no.
A common sin of mankind is that instead molding our desires to God's will, we try to mold God's will to our desires. My younger brother is an unfortunate example of this problem. Becasue he believes that it is merely his understanding of the nature of God (and believe me it bears little resemblance to Catholic or Protestant theology) that is the dividing line between Heaven and Hell, not any measure of absolute truth. I've tried to get him into any decent church of any denomination praying that he will hear the Gospel in a way that finally reaches him, but the harder I try, the more he resists (I - and he - could use great prayer on this topic. Am I planting a seed for someone else to water? Or am I damaging the cause of Christ because listing to his big brother is the last thing he wants to do?)
In Protestant theology, the "problem" is that you are free to define God's will in a way that suits your own will (see for example the splits in formerly solid denominations over the ordination of homosexuals, the role of women in the church, abortion etc. etc. etc.).
The "problem" for Catholics is that we must accept that God placed a structure within his church that can be relied upon (despite the sinfullness of many of it's members) to interpret Scripture corectly even when we don't agree with a particular decision.
I'll take door#2 any day.
No, sorry. There can only be one correct interpretation of the Bible. The problem I speak of is: "Is ours right"? The RC has exactly the same problem as any other "church" with this question, even though you fail to recognize it.
How old is he? Pray for him, but don't push him. God will find him. Sometimes we have to be stupid first. God's way of making sure the lessons we learn are particularly memorable!
I think that you, understandably, as an outsider looking at all Protestant churches see it much differently that it actually is. I do not feel free to define God's will in a way that suits my own, as you state. If this were true, that definition would be much different than what I believe is contained in God's Word. The real problem with these "churches" you referred to is not that they interpret God's will as they choose (though they surely do), but that they but are buildings filled(?) with people that aren't even Christians, they just play "dress up" on Sunday mornings. Therefore, of course, they don't believe or follow the Bible's true teachings, but substitute a counterfeit that makes themselves the god. As you might surmise, I am not an apologist on their behalf.