Steel road bike - is repair an issue?

If you're worried, have a proper old fashioned bike shop like Ellis Briggs take a look. They can do a proper repair if one is needed. Did a lovely paint job for me on an old frame last year and checked alignment, cleaned threads etc.

https://www.ellisbriggscycles.co.uk/resprays/frame-repairs/

If the drop out has been replaced and brazed, so long as it has been done properly, then I would have no issues. If the drop out has been welded dont touch it.

When you say "repaired (professionally welded) rear dropout" do you mean that the dropout was repaired or, as is normally the case, replaced by brazing in a new one. If a new dropout has been brazed in by a competent frame builder/repairer it should be as good as new. This is a common repair (or was when all frames were made of steel).

Other than the weld quality you also need to consider (as its a dropout) whether the spacing is correct and the alignment is good. Is it an integrated hanger, and if so is that the side that was welded?

Isn't this one of the the big selling point of steel.  You can actually repair it. 

I'd have no issue at all, again assuming the repair was carried out by a quality framebuilder. The work that Argos in Bristol do, for example,  is amazing and I'd have no issues whatsoever with treating a repaired frame by them to be as good as new.

Depends on the quality of the weld. If you respect the welder, its no issue. The rest of the frame is welded together so I don't see a particular problem. If its worth it you could always get it x-rayed by a local engineering company.