My ideas are as follows (in blue):
1. a. After getting ............, my parents made me take a walk with him every night.
I feel something is missing after 'getting'.
b. Will you have me raise the dog?
I feel the word beginning with 'l...' (instead of 'had') sounds better.
c. He let me get (have) a pet as I was good at my test.
I'd prefer to use 'I d.... well(good) on(in) my test.' rather than 'I was good at my test'.
2. a. The other monkeys were envious of ............ getting bananas.
Again, I feel something is missing in the '..........' part.
b. All of my friends are jealous of my dating Alec.
I feel this one is OK.
I might be wrong (if so, please forgive me...).
P.S. It would be advisable for you to start separate threads for separate questions.
Student or Learner