Bicycle Mechanics - Internal top tube reverse brake installation on bullhorn/tri bars

Long title, I did some searching (also browsed Sheldon) and couldn't quite find what I was looking for. If you have any links to past threads, please post them and I will gladly read through them to find my answers.

I'm installing brakes on a singlespeed build (my first bike build) and want to make sure I do it right and don't do anything that can't be undone.

As the title states, that is what I plan to do. Here are my questions

1) Is the cable naked through the length of the interior of the top tube? If so, how is the housing best precisely removed (accurately for length)?

2) Are there special ferrules that fit on the naked cable on the entrance and exit of the top tube?

3) I've attached pictures of my brake levers and the brake cables I've got. The ends don't seem correct for my needs. If they aren't what is the term/or a link for the cables I do need.

4) When I start, do i clip the bit of the actual cable that would attach to the brake assembly and remove the housing to be cut? I'm a bit confused on this part.

Any info/links/anecdotes are very welcome. This is the end of my build and I am very anxious to get it done with so I can finally ride it.

Thanks in advance!!

You seem to have a gear cable for internal gear hubs there.
You need a brake cable and housing of proper length.
Brake cables, at least generic and shimano cables, are double ended for both road and mtb brake levers. for that lever you need the MTB side.

the cable that goes through the top tube, depends on how it's made.
If the hole allows the entire cable housing to thread through, then you need to do it like that.
If there are cable stops that stop the housing from threading through the entire top tube, then you need to cut the housing to a proper length for the housing that needs to go from the bars to the top tube and the top tube to the rear brake.

Thanks for the info AEO. That should get me started!