We have two outside faucets on one end of our house that work fine. On the other end there is a faucet that hasn't worked since we've move in.

We have subsequently figured out that somewhere after the last visible location in the basement, that the pipe that goes to the kitchen sink and from there on to this outside faucet must have froze and burst, and the fix the previous owners implemented was to cut out and abandon that pipe, and run a new run of soft copper tubing to the kitchen sink. As shown in the following image:

(the little circles are valves)

Now that I'm redoing the Kitchen, including moving the sink several feet, and repairing the jagged holes they cut in the floor to repair/replace the pipes I'd like to see whether I can get the outside faucet working again. After doing the plumbing for the kitchen the current state of the pipes is as follows:

and I currently have access to the pipe right above the existing faucet (as shown) But I have a couple of questions

1) Is there anyway to determine whether the segment of the existing pipe between where I have access and the faucet about 4 feet below is still in good condition?

2) Since there is an existing soffit that contains two main ducts and much of the wiring for the house is it reasonable to fish a flexible pipe through this space? or does it need to be securely fastened every so many feet?

3) If fishing a pipe through is reasonable should it be a PEX pipe? Soft copper? (I have somewhat acidic water (and an acid neutralizer installed) which could make PEX a better solution, But I have also seen ample "evidence" that many mice have lived in the soffit space, which might suggest against PEX.))

4) If I go with PEX is there any inexpensive and reliable method for making 2 PEX connections? Id hate to have to buy a $150 dollar tool (or however much they are now) for making only 2 connections.