Explain voltage readings on NEMA 10-30 socket
In a 120/240V single split-phase system, there's two ungrounded (hot), one grounded (neutral), and in most cases one grounding conductor.  Measuring between the two ungrounded (hot) conductors should yield 240 volts +- 5%. Measuring between either hot and neutral, should read 120 volts +- 5%. 



In the case of a NEMA 10-30. Each angled contact is one of the ungrounded (hot) legs, while the "L" shaped contact is grounded (neutral).  Therefore, measuring between the angled slots should give you 240V. While measuring between either of the angled slots and the "L" shaped slot, should measure 120V.

As for plugging a NEMA 6-50 plug into a NEMA 10-30 receptacle, forget about it.  NEMA 10-30 recepracles do not have a grounding conductor, whereas a NEMA 6-50 requires a grounding conductor.   NEMA 10-30 is an nongrounded 120/240V device (hot, hot, neutral), NEMA 6-50 is a grounded 240V device (hot, hot, ground).  A 10-30 receptacle will be protected by a 30 ampere breaker, and supplied by wires only rated to carry 30 amperes of current.  If you plug a 50 ampere load, into a 30 ampere circuit. You will overload the circuit, and (hopefully) trip the breaker.

If you need a 50 ampere circuit for your welder, have a proper 50 ampere circuit installed by a licensed electrician.

You should NOT plug a 50 ampere device into a 30 ampere circuit!