Do doctors in the us offer testing for mycoplasma genitalium or is it only in the research settting? Will it show up on pap smear?

Not a routine test. Nucleic acid amplification tests (NAATs, e.g. PCR) for M. genitalium are now offered by many labs, but some tests are not FDA approved and M gen testing is not done routinely in STD checkups, except (as you suggest) for research. It doesn't show up on pap smear. Most infections are not believed to be harmful, so no worires if neither you nor partner(s) have discharge or other STD symptoms.
Culture. Usual test is culture but it is not very effective. Nucleic acid tests have not been approved by fda. If you think you have mycoplasma, you need to be tested for other sexually transmitted infections. Mycoplasma can be treated with antibiotics. Pap smear is not a good test for this infection.