Question: I am almost certain that the R in *regards* must be in lower case because it follows a conjunction and is therefore a part of the same phrase/sentence. But I've been seeing a lot of emails of late that end in "Thanks and **R**egards," which got me wondering.

Answer: 
Yes, many use that way, also in "Best Regards".

But, especially if we're talking about some official/formal email, I'd suggest to write according to the normal rules of orthography.

In that case, write them like this: "*Best regards*", "*Thanks and regards*" or "*Yours faithfully*", etc.
I think it would be stretching a point to rail against capitalising "Regards" in OP's example, though in most cases people only capitalise the first word.

I believe the intention here may be to convey *both* "single-word sign-off's" in a single phrase, where either on its own would obviously be capitalised. Though as @Alenanno points out, it's not uncommon to see "Best Regards" or ["Yours Faithfully"](http://www.lettersofnote.com/2010/06/yours-faithfully-adolf-hitler.html)anyway.

In short, it's a matter of style, not grammatical correctness.