How does 上 work as a verb?
I am a native speaker. For the example you made I feel some of them would be more authentic if you use other verbs instead of 上 for example: 
这本书我读了15页。 (From this book, I've read up to page 15.) 
我给自己剪头发了。 (I cut my own hair.)
(I stay in a hotel in Beijing for 3 months.) This one sounds a little bit weird in the tense itself. If you want to say "I am going to stay in a hotel in Beijing for 3 months" its a good time to use 上，it would be:我要在北京一家旅馆住上三个月。 However, if you want to say "I have lived in a hotel in Beijing for 3 months", it would be: 我在北京一家旅馆住了三个月。I know it is to consider tense in Mandarin since it doesn't have a very specific rule for tense. 
我老婆拿了遥控器。 (My wife took the remote control.)
Here are some phrases that are commonly used with 上 as a verb:
我去上厕所 (I am going to the bathroom)
上床睡觉 (go to bed)
我上车了(I just got on the car) 
我要上台了 （I am about to get on the stage) 
上楼梯 （go upstairs) -> 下楼梯 （go downstairs)
I hope these could be somehow helpful.