I can agree with there being a first cause, but I don't see the logical conclusion in it being a god responsible for the first cause out of all other possibilities. That's a non sequitur. And if you claim it to be a god, than it is a GothG argument -> argument from ignorance.  And if you go deeper into the rabbit hole you'll end up on a special pleading. 

I understand your position. I agree the strongest criticism against the Cosmological argument is that: does God really follow from it? But I maintain that the GotG argument and special pleading aren't properly aplicable here.

Why not? When someone is placing god at the end of the equation it's because of ignorance. 

We don't know what caused the Big Bang
- I do, it was god. -> god of the gaps

We don't know how life started on this planet
- I do, it was god. -> god of the gaps

We don't know why there is more matter left in our universe. Matter and Anti-Matter should've all cancel each other out.
- I do, it was god. -> god of the gaps

 Using god of the gaps only for its original meaning / purpose would be a genetic fallacy. 

No, when people place God at the end because of ignorance, you have a God of the Gapps.

When you construct an argument that leads to this conclusion, it's obvious this doesn't apply.

1. The argument isn't debunked. It receives criticism, but it is quite sturdy.

2. Your presentation of the Cosmological arguments is too simplistic.

A good short explanation would be (and note this is only one abridged form of the cosmological argument, there are many others):

1) Everything is caused and everything that causes is caused itself (these are contingent beings and events, who can cause and are caused).

2) These events of cause and effect happen in a chain (one is caused and causes further).

3) We can follow this chain backwards in time, since in the future, it goes on for eternity.

4) To start the chain, in the beginning, there must be a necessary being. One that causes but is naut caused itself.

5) This necessary being is necessary since if the chain could not begin with a contingent being or event (since it cannot cause itself)

6) This necessary being is God.

I can see people being critical of the jump between 5 and 6, but as you can see the argument is quite a bit more refined then what originally stated. The problem is many lay people have a wrong conception of the cosmological argument and attack it with the wrong arguments. The argument is still well debated up to this day.

Yes.

If you've studied philosophy, you're probably aware that when you remove God from the equation epistemology and morality break down. A lot of people disregard this because they think they can know things and be personally moral without God, but the shared social framework holding criminal justice and academic discipline breaks down. This is exactly why Postmodernism is a thing and why it came about shortly after Godel's theorems of incompleteness proved mathematics itself was unprovable within its own assumptions.

EDIT: I suspect this lack of shared social framework is also part of the reason the Left has become far more vocal and shrill as of late. It doesn't have any actual moral high ground, so the major way they pressure people is via the bandwagon argument. 99% of scientists, etc. When that illusion starts to break down the only path forward is to abuse people into submission.

If you dislike religion because it tends towards orthodoxy, take note; academia and the agnostic left are going down the same path because orthodoxy is a result of the human condition and not a problem unique to organized religion.

Why do I even study Moral Sciences when everyone seems to know philosophy so well?...

Bot hare not applicable in this metaphysical argument and I've explained this enough several times over on this thread. Moving on...

Oh, morality can only come from a god? That's not true at all. Take some things into consideration:

1. Morality is not exclusive to humans. Read this article. Other primates were shown to also have a sense of morality, and we don't see them reading the bible or praying, do we? Primates have morality because it helps to reproduce and move on your genes when you don't go around killing everyone from your group. Sharing food makes it more probable that more members of your group will survive, trying to save a member of your group (and saving it) will make it more likely that you can pass on your genes. Moral genes were passed on because it helps the species survive.

2. Morality is subjective. If morality really did come from a god, everyone would have the same morality, right? So how do you explain slavery being a moral thing until a few centuries ago? How do you explain possession of women being moral until a few decades ago (and still being moral in some cultures)?

But it doesn't, at least not by logic. 

What is the logical justification to jump from first cause to god. 

There are different depictions of the cosmological argument, which the simplest is this one: 

1) Everything that exists has a cause of its existence.
(2) The universe exists.
Therefore:
(3) The universe has a cause of its existence.
(4) If the universe has a cause of its existence, then that cause is God.
Therefore:
(5) God exists.  

How do you jump from 3 to 4? 

Even if this is just how the argument is being presented, I still don't get how they came to this conclusion. The only thing is being bias towards god's existence and being ignorant of any other options during it's creation. I know this argument has been already debunked and the criticism it gets is in my eyes also justified. But still.. how do you make such a jump?

I see it the same as that one from South Park: 

Stealing socks
???
Profit

And everyone is ok with it.