You need fresh rosemary and give it a quick char and not allow it to burn. That's sort of the same technique of using a blowtorch for crème brulée and the same "doneness" of charred vegetables - caramelized but not burnt
If you're smelling smoke, it is already too burnt to use.
Yes, it's safe.
Silver tarnishes easily, particularly in contact with sulfur-containing foods, particularly egg. The tarnish is not hazardous, and can be cleaned off with silver polish or with DIY chemistry like in Tetsujin's answer. Black tarnish doesn't form readily on its own, so if you're not planning on using eggs in your cocktails then cleaning off ...
Cheap & cheerful trick...
Break your shaker down into its separate components.
Use your washing-up bowl.
Get 30cm or so of aluminium kitchen foil, & get it to sit in the bottom of the bowl. Doesn't matter of you have to crinkle it but don't screw it into a ball.
Drop in your silver.
Add half a packet or so of bicarbonate of soda.
Fill with very hot ...