Electrical – Proper way to route new wire through existing exterior wall

I'd trust the old reliable electrical tape over the liquid stuff.

Section 334.116 of the NEC (USA National Electric Code) talks about the sheath in very generic terms: it has to be flame retardant and moisture resistant, etc. There is nothing particularly special or magic about the requirements.

I'd inspect the wire very very carefully, maybe even remove the paper to ensure nothing is going on below, wrap it, then mention and show it to the local inspector when your job is inspected. Maybe even write a note on the cable in sharpie.

The worst case is you have to splice the cable into a junction box that forever more must be accessible.

The best way to deal with this, since you have the walls open, is to turn off the power and then find the nearby electrical boxes where these wires terminate and un-string the wires back out through your framing area. After new framing is in place then drill new holes for the wires and re-string them back to their destinations.

In some instances these wires will just be passing by the area where the walls are open but not terminate in any boxes in the open wall area. In this case life gets all that much more complicated. There are several strategies to deal with the latter, none of which are particularly good ones.

1 - You can notch the new studs to let the wires pass by. This requires extra care to protect the wires including the placement of nail / screw stopper plates over the notches.

2 - Cut the wires near the center. Then you could choose to add some new outlets to that wall and use the electrical boxes behind the outlets as splice points to support adding the new length of wiring needed to bridge the circuits back together.