Translation:I will buy a watch.
All clocks are tokei - wall clocks, digital clocks, watches, etc. Udedokei are only wristwatches. Ude literally means arm, so it's an 'arm clock,' or wristwatch. (Incidentally, 'wristwatch' should be accepted, not just 'watch')
Yes, especially since "watch" could refer to a pocket watch, which 腕時計 cannot refer to, making wristwatch a more accurate translation.
the sentences seem very weirdly structured. Why is it not I will buy a watch?
That answer is accepted. Japanese doesn't have a future tense, so the -masu form can be translated as the present or future tense.