yes, you should be able to check bags through.

separate tickets are a very bad idea. if you are late, the airline has no requirement to honor that ticket. LH probably has a lenient policy, but if you are not there when the flight leaves, the problem is yours, not theirs. did the person selling these tickets tell you of the problems?

FRA is a miserable big airport, and a 90 min connection time should be expected. yes, one can connect in 45min on some flights, but from one end of that airport to the other with an immigration and a "tsa" type inspection, 90 min can be tight, particularly if the tsa types want to keep one of your items(they wanted to keep my tooth cleaning machine)