I don't understand. You seem to be showing coils of wire on a workbench. The question is what is the wire IN THE WALLS between the breaker box and the dryer outlet. That is what counts, not some extra wire you have sitting around.
This is not a shock hazard situation. It is a FIRE hazard. If the neutral conductor is too small, it can heat up and start a FIRE inside the walls. You don't want that. There could be a delayed reaction. It may work OK for minutes, hours, days, weeks, months, even years and then, ONE DAY you are standing on the curb looking at the smoking ashes that the fire trucks are hosing down. And wondering if your insurance payments are up to date.
You obviously do not know the issues involved. If you want to do this, get a qualified electrician to inspect the wiring. But I would just have a new circuit installed for the welder. Money well spent!
is the wire size for the neutral smaller than for the hots?
It's standard 10/3 electrical wire. Pics attached.
The black 10/3 wire is from an old dryer. The orange is new 10/3 romex.