No. I'd be very surprised if grammar rules ever require a font or style change. It may be a convention for the publisher of your paper. Publishers publish style guides. Look for it.

There is no need to italicize. You can simply capitalize the first letter of the brand name, like you did in the first word of your example.

You should not italicize the brand name. But you should capitalize it. If possible, you should write the brand name exactly as the company does. So in this case: Lancôme. If you don't know how to type the "o" with the mark over it, you can do like I just did: copy and paste the name from the company's website. (Sometimes it's awkward to write it as the company does. "italki" for example is written with a lower-case "i" so if you begin a sentence with "italki" then it looks awkward. In cases like that, you should use your best judgement).