Are there special protocols one should take when handling vintage/old electronics?
A loose list. I'm torn between this question being far too broad and far too interesting.
What I shall clearly not do is address the physiological effects of chemical compounds. Just so much: don't downplay the dangers of things like lead poisoning. It's dangerous. Keep your hands clean, don't eat while handling equipment, standard etiquette.
I can't also give you good handling advice. In fact, it's simple: Old electronics can contain hazardous material. There's national standards on how such must be handled. Don't settle for less &quot;just because you're a hobbyist&quot;, &quot;small entrepreneur&quot; or &quot;will be doing this for only a short time&quot;.

Mercury and Cadmium batteries, old batteries tend too &quot;bleed&quot; those
Lead migrated from solder joints, contained in particles (&quot;dust&quot;)
Bromine-based flame retardants permeating everything, especially printed circuit boards in the pre-glass fibre era
Asbestos in the isolation of temperature-controlled components
Chromium in lacquers
DEHP, BBP, DBP, DIBP as abundant plasticizers in polymers
Boric acid in electrolytes of capacitors
Fire hazards due to aged capacitors
Electrocution hazard due to aged cable isolation, esp. in transformers
Microwave equipment, esp. gyrotrons and the similar contain very hazardous alloy components
Fire and electrocution hazard due to conductive or hygroscopic dirt
accumulations

Device type-specific dangers of common appliances:

CRT TVs contain high voltage supplies
damaged CRTs can implode, scattering very sharp glass shrapnell. Security of old Cathode Ray Tubes is not up to modern standards.
CRTs emit X-Ray radiation. Shielding only in the front. Old TVs are not up to modern standards and will cook your organs
Tube amplifiers have 500 V+ as acceleration voltages
high-powered transformers are commonly suspended in oil, which has been heavily treated with very poisonous chemicals.
Old fire warning systems contain radioactive material in significant amounts
Old gas discharge tubes may contain radioactive gas.

International references
a good idea is to look into the RoHS (Reduction of Hazardous Substances) legislation and the laws it build upon. That gives you a good idea of what used to be abundant.