I want to run 3 circuits: 120v using #12 thhn for washing machine on 20 amp breaker, 240v using #10 thhn for dryer on 30 amp breaker and 240v using #6 thhn for stove on 50 amp breaker.
Can they all be run in the same emt conduit and then separated in a common j box? What size conduit is needed for all three circuits and what size ground if required?
Yes, it's allowable to run multiple line voltage circuits of varying amperage's thru the same conduit. No low voltage circuits (eg: Coax, cat-5, cat-6,etc) can share a conduit with line voltage circuits.
Pay attention to conduit fill capacities as it can get complicated with varying gauge wires. Going larger never hurts and makes for easier pulls.
Normal rules for conduit apply: no more than 360 degrees of bends without a pull box: A box that the wires run thru but no connections. Your proposed split out box may help meet that requirement. Like crip said, if all metal (boxes and conduit) can serve as a ground, but it never hurts to run an actual ground wire.
EDIT: South Wire has a calculator that help determine the right conduit size. I just ran it with your specs and it comes up with 1" conduit. Personally I think that's too small, but that's just me. I'd go with 1.5" conduit to be safe.https://www.southwire.com/calculator-conduit
There are two separate issues - conduit fill and multiple conductor conduit derating.
Conduit fill is easy. There are plenty of calculators. For example, using the Southwire Conduit Fill Calculator and adding 2 x 12 AWG, 3 x 10 AWG and 3 x 6 AWG, 1" conduit will work (and 3/4" will not). Even if you add a ground wire (8 AWG is sufficient), 1" will still be fine. With EMT, you don't need a separate ground wire, with PVC you do.
The derating is a bit complicated. The details are in NEC table 310.15(B)(2)(a). You have 6 conductors (the neutrals don't count for this), so the wires are derated at 80%. The starting point depends on the type of wire. THHN is normally rated at 90 C. 12 AWG = 30A x 0.8 = 24A, 10 AWG = 40A x 0.8 = 32A, 6 AWG = 75A x 0.8 = 60A. Since these derated values are all larger than the normal breaker size (which factors other things into the equation), there is no effect from the conduit multiple conductor derate. Note however that an additional circuit would change to a 70% derate, and that's where it starts to become an issue. 10 or more wires (not counting neutrals, so basically 5 circuits or more) and the derate goes to 50%, which effectively means that it is usually going to be more practical to simply have a second conduit if you have more than 4 circuits.
