If you have a felony conviction within the past 10 yrs, you can be categorically disqualified from getting licensed. It is not a sure thing tho as it depends on the conviction, in this case, credit card fraud which is fiduciary by nature and one of those things that is looked down upon. If the conviction is greater than 10 yrs, it would not be (much of) an issue but would still have to be listed when filling a U4. There's a lot of misinformation about what is and isn't so I suggest you spend a little time and Google the SEC act of 1934 (section 15(b)-4(b) to be precise) and read. If still not convinced, ask your compliance officer - that's why they are there. (I had a skeleton or two and have researched it before I got my own license. It will just always cause extra scrutiny, but so long as I keep doing the right thing, I will not be denied a license.) Bottom line, if a firm is willing to take a chance on an individual as such, then the responsibility falls on the firm to monitor the behavior of the individual and they will be held accountable.
you can trade anything, not just fx few years back, i killed a hooker when i was on a coke binge (it was ok, i left her a big tip). didnt stop me opening an account with mfg. in fact, they didnt even ask! how lucky is that!