with our compliments
1. I will send you the 1st issue with our compliments.

Here, What does "with our compliments" mean?

2. Nicholas plays Steven Stelfox in the movie, which follows the businessman as he slashes his way through the music industry to secure his career. It's branded a dark comedy, a genre, the 25-year-old certainly took in his stride during shooting.

Here, doesn 'follow' mean 'admire'?, Is "as" used as the meaning of "like"?

Thanks in advance :)

Question

with our compliments
1. I will send you the 1st issue with our compliments.

Here, What does "with our compliments" mean?

2. Nicholas plays Steven Stelfox in the movie, which follows the businessman as he slashes his way through the music industry to secure his career. It's branded a dark comedy, a genre, the 25-year-old certainly took in his stride during shooting.

Here, doesn 'follow' mean 'admire'?,  Is "as"  used as the meaning of "like"?

Thanks in advance :)

Answer

1) The phrase "with our compliments" means something is free. You don't have to pay for the 1st issue. Also, the word complimentary has the same meaning. Complimentary drinks = free drinks

2) The word "which" does not refer to the person, but the movie.

Admire is not the best synonym for "follow" in this example. Admire has the added meaning of really liking something, maybe becoming the object. 
"Follow" in this example means to watch something closely (track something, like an animal). Liking or disliking the object is not important.

"As" is used as a conjunction. It used like "when" or "while".