It is a German borrowing. Russian has a number of everyday words that have actually been borrowed from other languages. Most of them are not from English: by and large, only the very recent loanwords come from English (most of them related to technology, business and entertainment).
Do some words not come from Old Norse? This was technically a Germanic language anyway though. Was just reading about the twisted paths between Slavic and Germanic languages some words have taken to get to modern Russian and German, such as the German word, knute, which came from Russian кнут, from Old East Slavic кнуть, which in turn came from Old Norse knútr, and I believe all of which mean something like whip or knotted cord.
Bare У меня does not focus on the fact you really HAVE that thing, just specifying what it is. It is assumed that either it was obvious you have at least some kind of that thing ("I have dark hair")—or that the fact of possession is not as relevant right now as the identity of the object ("He's got a gun!")
The presense of есть (which is,effectively, "is") explicitly says that such an object exists. As such, it is not used to describe properties or quantities, even though in English "I have" can be easily used for these as well. For example, in "I have 4 sons" it is very likely that the number is the focus. In "I have dark hair" it is a given that you are either bald or do have some hair at least—so «есть» is not used in such situations.
Есть is also dropped when talking about illnesses ("I have an influenza/diarrhea/allergy...")
So, if my wife is cooking eggs and says, "I need something to put these on!" and I have a plate sitting in front of me, I'd said «У меня есть тарелка» - I would not omit есть,
On the other hand (OTOH), perhaps I am at a meeting about an upcoming lunch, and someone says, "We will need 200 plates.", so I answer not-so-helpfully, «У меня тарелка!»
"I have the plate" would be "тарелка у меня" without "есть". "Есть" is used when we are establishing the existence of something, while when we are talking about the plate it's existence is not in question.
"Тарелка есть у меня" is just an unusual word order, but it still means "I have a plate". I'd translate it as something like "I, for one, have a plate".
Тарелка is something you eat from.
Пластина is a thin object made of rigid material
плита is a huge stone or metal slab. By analogy, it also means a range (the one you have in the kitchen). The only meaning where I think it can translate to "plate" is if you mean a tectonic plate or, I don't know, a layer of rock.
License plates are none of the above; they are номерные знаки or simply номера ("numbers").
I hear tarelka. It's being pronounced correctly. You can also hear it pronounced here: http://forvo.com/word/%D1%82%D0%B0%D1%80%D0%B5%D0%BB%D0%BA%D0%B0/