From SLC, one could take I-15 to I-40 then cross over to UT24 to Hanksville, but it is definitely much longer of a trip. US-6 from Spanish Fork to I-40 is four lane for a good portion of the drive to Soldier Summit, and is 65mph the whole way except near Price.
I-15/I-40/UT-24 : approximately 369 miles
I-15/US-6/I-40/UT-24 : approximately 231 miles
There are plenty of other variations to these too if one wanted to mix it up even more.
I really can't give you an update on the snow in that area but I can assume that the area did get some new snow after the storms that have passed through the state in the past week. The only person I know that has hiked Mt. Ellen was Erin (Wired) back on April 7th. Here is a link to her fascinating trip report: http://www.walkingwithwired.com/2015/04/day-16-mt-ellen-in-wind.html
You could also check with the BLM office located in Hanksville.