Multiple circuits in conduit is not a problem
OK, so you want the details. Yes, you can run multiple circuits in conduit if:
- they meet physical conduit fill rules, which is easy with THHN/THWN wire at your sizes
- the wires are electrically derated, which is irrelevant below 3-4 circuits at these sizes.
One ground can be shared amongst all circuits, it must be large enough for the largest circuits. With EMT, IMC or rigid conduit, the conduit is the ground and no wire is needed.
For the physical fill, just use a calc, and you use it on the actual physical wires you have in the pipe. For you, that's 2 #12 and 4 #10 unless you're using pipe as the ground, in which case only 3 #10. Any conduit size will handle that. 3/4" will give easier pulls. But if you use stranded THHN, the pulls won't be bad at all.
For the electrical derating, in residential, don't worry about it if have 3 circuits 60A or smaller. Also don't worry about if you have 4 circuits in 15/20A.
Oh, you want the gory details:
OK. We start by counting conductors in the conduit. Here's the important part: Grounds do not count. In 120/240V split circuits with 2 hots (including MWBCs), neutrals don't count either. So here is your list of wires, and what counts and what doesn't: 120-Hot 120-Neutral 240-Hot1 240-Neutral 240-Hot2 Shared Ground ... Total 4 wires.
Then proceed as in my answer here.
Table 310.15b3a says 80% derate (don't panic) then 310.15B16 says #12=30A=derate 24A. Well, that makes no difference, since we're already limited to 20A. #10=40A=derate 32A. Again no difference. This logic applies up to 3 circuits per conduit. At 4 circuits (8 countable wires) we need a 30% derate, which doesn't bother #12 or #14 wire. At 5-10 circuits (10-20 wires) we hit a 50% derate, which means bumping at least 2 numerical wire sizes - no fun.
On the dryer's ground strap
You must follow the labeling and instructions. Modern dryers, when wired for a 3-wire connection, instruct you to add a bootlegging jumper on the dryer. Either this procedure will imply how to "roll it back" to 4-prong, or the instructions will explicitly say. There is no substitute for following them.
The 3-prong configuration is dangerous because if the neutral wire breaks, the dryer chassis is energized. If you are stuck at 3-prong and can't retrofit ground, the safer thing is convert the dryer and receptacle to 4-prong anyway and fit a GFCI breaker. Label the receptacle "No Equipment Ground".