What type of material is my countertop made of and how should I clean/maintain it?
It is going to hard to tell precisely what type of surface you have. Too many man made look a likes out there.

But to guess, and that is all it will be, that it is a flamed surface granite. I have seen it used in lieu of natural cleft slate which has a great look, but terrible for counter tops. Granite could be sealed, but it does not take it very well, it is just really dense. If it is Quartz, then typically that is man made and does not need to be sealed. The resins bonding the quartz together provides that function.

I have seen, when sealer is applied too heavily and not wiped off well enough, that it will turn white from moisture, otherwise where untreated stone or other material will not. I would consider using a stripper of some sort for your sealer you are using, let it set and see if the white goes away, if you can get it down to bare stone again. 

The area around the faucets is my clue as to what is going on, since those tight spaces are hard to access for getting excess sealer up.

As a mention, if it is Corian, it definitely does not need sealer. Nevamar is a Corian competitor, same thing. Silestone and Ceasarstone among many others, are quartz providers. So if you get a chance to look in your records, if you can find them when they installed, will give you and us a better idea on how to proceed.

And yet another mention, about the chip on the corner, there are epoxies specific for these types of tops, as well as colorants to make that chip disappear. If you got a stone top installer over for a visit, they could fix that for you in a jiff. Besides tell you what the material is that you have...