Is it safe and feasible to use a "damaged" transformer?
The most important thing to check for safety is the insulation on the primary winding wire. Any contact between the wire and the core could cause the core to be at the primary voltage level. Connecting the primary only through a ground fault interrupting (GFI) receptacle or circuit breaker is essential. A connection between one primary winding and another would be a shorted-turns situation. That could cause a fire or damage that would result in a connection between the winding and the core.

Damaged lamination insulation will cause increased heating. The transformer could get hot enough to damage the windings. However, that would probably not happen very fast, so it would probably ok to use the transformer if you never leave it operating unattended. Unless you have a great deal of experience, you should never leave any home made electrical project operating unattended.

Loose laminations will vibrate and make noise, The vibrations might ultimately damage the insulation between laminations.

At welding voltage levels, a damaged secondary winding is of less concern, but a shorted turn would cause the winding to get hot pretty quickly.