Unless you're living in some place with bizarre code restrictions you can just replace it all with NM. However, if you want/need armored cable, type AC and MC are both available readily (Lowe's here even has it). You definitely need to get rid of the "flying splice" either by replacing the cut cable or moving it into a box.

I can't follow some of your other questions or description of the problem. "old stuff from stove below" means what? If your hood is tapped off the power for the stove, this in itself is a violation even before people started tapping off microwaves.

You don't need to have a junction box if you can bring the cable directly to the fitting on the hood. If you do need a box, then yes, you need to use an appropriate fitting. In fact, you may need to replace the fitting on the hood to match whatever cable you end up using.

Its gotten a little bit more complicated

I realized that the stove and the hood (both on same 20 am p circuit) are wired with 14/2. The armored cable leaves the top of the box and then on the otherside of stud is connected to the NM with a flying splice. There is no slack to pull it up to the hood.

Too top it off, there is 1/8 or so thick paneling surrounding the outlet behind the stove. I am not sure how to cut it to access the top of the box where the severed armored cable is connected too.

Perhaps I have to bit the bullet and call a sparky.

This is a gas stove? 20A won't power an electric one. You are going to need an accessible box or completely replace the wire going to the hood. If you can't figure out how to do this, then yes, call in the pros.