Eh, leaded solder is just fine with some basic precautions (and still indicated for military and space ratings) but the smell is probably a highly reactive core flux (rosin) that will eat away at anything it's used on and needs to be vigorously cleaned off after soldering. It may also be a sensitizing allergen so maybe kabdib shouldn't use it, at least not without an extractor.

If you're doing more than the once a year small project you should have fume extraction, occupational (or hobby-based) asthma sucks. As does metal flu (zinc for welding and tin for soldering although I expect that's rarer).