This is not really a continous construction in Dutch, although it does look like one. Here "het middageten" is a noun, which is combined with the preposition "aan" to describe where we are (figuratively) sitting. One could also say "Wij zijn aan het lunchen", which is a continous construction, using "aan het" + infinitive.
Well yes, but your example uses te + infinitive, making it indeed a continuous construction. That is not what is happening here, and the idea of "half a construction" doesn't really make sense to me.
What about "Wij zitten aan tafel" = "We are sitting at the table"
Would that be appropriate for Prepostitions? I'm my opinion there is not much of a difference with this sentence.
From what I've gathered, the fact that they sound the same when spoken fast is how the unstressed (we/ze/je) forms came into being. When you're speaking fast — i.e., not emphasising the word — jij sounds like je, wij sounds like we, and zij sounds like ze. Thus they kinda became their own thing.
Yep, you're right. They don't even just sound like je/we/ze, but that's what we actually say when we're not stressing them. You mostly use jij/wij/zij (note: hij does not change in he) as a stressed form or in a more formal text, but they're not wrong in informal situations or when you don't want to stress them.
No, because Dutch is more regular than English in this respect. It's the day's uniquely determined lunch, so you would expect it to get a definite article. If you think about it, it is odd that in English we omit the definite article for activities that happen regularly every day.
This is a very literal translation though, which wouldn't be idiomatic English.
I can think of a context where there would be a definite article in English. Where a particular lunch, dinner is being referred to. For example: 'Are you going to the dinner after the conference?'.
The use of articles is one of those areas where languages tend to be highly idiosyncratic, as is preposition use. It's not uncommon for something that sounds completely wrong in one language to be the rule in another. There generally isn't much logic behind it either - you pretty much have to just learn these things by rote.
No, this has already come up several times on this discussion page.
Sit down and to both imply a direction, so you can only say your English sentence at the beginning of lunch. Aan translates to at, by, so it is completely static. Most likely lunch is already in full progress, and in fact it may be nearly over.