"Thanks and Regards" vs. "Thanks and regards"

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"anyway.

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

If rendered this way: "Best Regards," then you have an example of honorific capitals. This is an error committed by those who do not understand the rules of written English. Of course it should be "Best regards," as only the first word of a sentence (and any proper nouns) should be capitalized.

But some people don't know that, and instead have internalized a false rule along the lines of "capitalize all important words." This is quite common in internal business communications.

For example, in a company newsletter, you might see something like this: "The President of Balco, Claude Balzer, will be speaking to the Head of the Engineering Department." In this case, only the 'T' in the first "the," the 'B' in "Balco," and the 'C' and 'B' in Balzer's name should be capitalized.

In a similar way, the writer of "Best Regards" thinks that the complimentary close is a special part of the letter or email, and thus worthy of separate rules that do not apply to any other part. There is, however, no uniform style manual that authorizes such a practice.