Learn Dutch: Ik kies dat horloge. - I choose that watch.

Because horloge (watch) is a het-word (het horloge). In Dutch, you must use dat (that) or dit (this) to point out a singular het-word.

If I wanted to ask "Which watch?", would I use welk or welke?

You would use welk horloge. The rules for "which" exactly match the rules for "this/that". Because horloge is a het-word (which is why we say dat horloge), you use welk for the singular. You would ask: Welk horloge kies jij?

What happens to the verb kies if I want to say "he chooses" instead of "I choose"?

The full dictionary verb is kiezen (to choose). For ik (I), we use the basic stem kies. For most other singular subjects like hij (he), zij (she), or jij (you), you add a -t to the stem. So "he chooses" becomes hij kiest.

How do you pronounce horloge? Does the "g" sound like a normal Dutch "g"?

No, horloge is a borrowed French word. Instead of the typical hard, scrape-in-the-throat Dutch "g", the "g" here sounds like the "s" in the English word "measure" or "treasure".