RNA is a single-stranded nucleic acid and tends to form secondary structures. So, a standard agarose gel will not give an accurate size separation of your RNA sample. Therefore a denaturing agent like formaldehyde must to be added to the agarose gel and the RNA to ensure the molecules remain single-stranded.
It depends upon the downstream uses. In case the downstream use include a sensitive experimentation, then yes you should run MOPS gel or denaturing gel or bioanalyzer. Otherwise TAE will do.
There should not be any contamination of RNase in any case. Be careful.