Can indoor cable be connected to outdoor cable of the same gauge

The purpose of the extension on a frost free faucet is to get the valve itself in a heated space inside of your home so that it could never freeze (see the below diagram). If you only have an inch or so of insulation that's being penetrated by these plumbing pipes, it's likely that the valve would still reach the freezing point inside of your wall. So any type of build out on the outside of your home to mount the faucet is defeating the purpose of the extension. The proper way to use this type of faucet would be to build out something on the inside of your home, which I'm guessing isn't possible, or at least unattractive in your situation.

Therefore, I'd recommend getting in the habit of shutting off the water on the inside, possibly replacing the shutoff valve with a quarter turn ball valve so that it's quick, easy, and obvious when it's been done. You want to open the outside faucets when the water has been shut off so that the lines are empty when they do eventually freeze. After they've been left to drain for a day, I cover my outdoor faucets with insulating covers. This is also a good time to drain the hoses and bring them into a protected location (e.g. a garage or shed).

Your UF cable can be easily spliced to either add in a new piece of cable or to rejoin a break in an existing cable. The splice kits are typically specified for copper wire only and consist of some set screw cylinders that join the wire ends and then some heat shrink tubing that has a very sticky adhesive inside. The splices can be buried and it may be preferable to do so from an appearance standpoint so just new wire shows above the ground level.

One product I am aware of is from 3M company and is called their "UF Splice Kit" as shown below.

Another very similar product from Tyco is available at Home Depot stores.

Do read the customer comments on the Home Depot page. One thing to learn from there is that a torch is not right tool to use for shrinking the tubing. Instead I recommend using an electrically heated hot air gun. (Although do use care using an electrical appliance outside, particularly of you are working in a wet hole in the ground). Another point of note is that successful sealing of the UF cable with these splice kits is to clean the wires very well before applying the tubing.