Re 1850, just what exactly would Fillmore's critics prefer him to have done?
Taylor's proposal, to admit CA and NM immediately as free states, might have got through the House, but stood no chance in the Senate, where Democrats and Southern Whigs would have combined against it. So if Fillmore vetoes the Compromise measures, all that does is to leave matters on hold until at least 1851, and probably 1853. Since Northern and Southern Whigs would be at each other's throats, the 1852 election will be much the same sort of Democrat landslide as OTL, whereupon the Compromise measures will be signed into law by President Pierce, with any changes more likely to be in favour of the South.
As it was, the Compromise was pretty much a northern victory, with CA admitted at once without slavery, and NM and UT likely to follow. All the South got in return was a Fugitive Slave Law which it would have got anyway a couple of years later, and putting off the evil day for a few years in regard to admission of other free states. Fillmore would have been crazy to veto the measures.
As for the "Mormon question", afaik Utah gave no trouble during the Fillmore Administration, or even the Pierce one. As I understand it, Brigham Young's appointment as Governor received Senate confirmation without a debate, which would seem to indicate that it wasn't seen as controversial at the time.