Can I transit through Bangkok without a visa if I have a separate onward flight?

As Brad pointed out, most of the low cost airlines fly out of Don Muang airport (DMK). So if you do plan to have an onward flight, make sure it is the same airport. Bangkok Airways would be a good one to check as it flies out of BKK.

If you only have a ticket to Thailand and need a visa, the airline will not let you fly.

If you have two tickets Bangok/Yagon you can transit through Bangkok airside (without going through customs) if you are not issued a second boarding pass in Shanghai there will be ticket desk in transit where can get it.

Check carefully your requirements for Yangon, its not the easiest country to get into.

As regards your luggage I'm not sure, but suspect there will be a place to stow at the airport.

What about an Exempt Entry into Thailand 30 days - no visa needed.

Even separate booked tickets from different airlines, sometimes airlines allow you to do thru-chekin. Or if you can on-line checkin in advance, maybe you just need to go thru to "international transit gate".

I think you can contact both of airlines to make sure. And contact to their office in your country AND Bangkok office.

I have experience of my friends got similar (but transfer at BKK to Thailand domestic destination) situation. When they asked to the airline office in their country about thru checkin, airline said "cannot". But when I asked to same airline at Bangkok office, they said "You can". I asked too detail procedure in Bangkok airport, and have them assured that my friends could. And my friends were happy to travel easier. This was just three months ago.

I think you woukd near to clear with all airlines before leaving shanghai otherwise you would not be able to get a boarding pass. Unless you booked well in advance and checked in online. I know a lot of local airlines wont let you book online if its 2 hrs or less before the actual flight.