The answer is:
From me to the Persian there's many a way
There's more than one way to skin a cat, Persian or otherwise.
But then, speed has family from teeth, so they say.
S (speed? In technical contexts, it's more commonly v for velocity) + KIN (family). And you can escape or win by the skin of your teeth, that is, only just.
The depth that a Lovely may well be found in
"Beauty is only skin-deep."
A treatment for goiter that's found in a tin.
KI (potassium iodide) inside SN (tin). A cause for goitres is lack of iodine and a remedy is to use iodized salt, of which KI is a possible ingredient.
Now take back the coin, by the thousands it falls.
NI (nickel, a coin and a chemical element) + KS (thousands), all reversed
To slide– what a gas for the first coveralls.
SKI(d) with its trailing letter removed as indicated by the dash + N (nitrogen, a gas). And, of course, the skin is everyone's first coverall.
The sniffer is the nose and "That's no skin off my nose" means that you don't care deeply about something. If "skin" isn't the right answer to this riddle that's no skin off my nose.