Well done with these sentences! :) There were a few corrections needed, I've put ( ) around the corrections. See below :)
1. My boss had (offered) me to take a one-week trip to Los Angeles.
2. The new (plants-no 'e') have enriched the fauna of our area.
3. To mark the painting she has carved a name on top of it. (This is technically correct but I'm wondering if you meant 'sign' instead of 'mark', if so 'carved' would also change. If not, awesome work!)
4. He was one of the heirs (of/to - to is more common) the THRONE.
5. It was raining for three consecutive days. :D awesome
6. We made an agreement that I'll come to the office at 6 o'clock (in) the morning so I can pick up the necessary documents for (my/his/her/their) visa.
7. He urged his friend to go (to dinner) to one of the most (fashionable/popular) restaurant(s) in the city.
8. A shark is one of the aquatic animal(s). ('A shark in an aquatic animal.' would be more common.)
9. I have never seen such (a) strange, bizarre thing.
10. I love drawing on coarse paper. :)
11. Fruit digests in our organs(stomach) in a few hours.
I hope this helped. :) Your sentences are great.