The first three sentences all end with prepositions. Although the Grammar Doctor thinks the rule about prepositions at the end of the sentence should be revoked, in more formal English it is best to avoid prepositions at the end of the sentence. (See the Grammar Doctor's view on this: prepositions)
The first sentence should use " whom" rather than " who" since it is the object of a preposition.
(To whom should I send this?)
4. He began to really try hard.
5. He began to slowly realize what had happened.
Sentences four and five have split infinitives. (Horrors!)That is, they put an adverb between " to" and the rest of the infinitive. This is a violation of Latin grammar, so some eighteenth century pedant decided it ought to be against the law in English grammar as well. Because we still have plenty of pedants around, we have to avoid splitting infinitives in more formal language.
(He really began to try hard.)
6. He's usually easy going. But before his first cup of coffee, he's a bear.
7. This department is going to work together as a team. And don't you forget it!
Numbers six and seven contain sentences that start with conjunctions (" but" and " and"). You remember your English teacher telling you not to do that, don't you? It should be avoided in more formal English.
8. Are you sure it's finished? Checked every detail?
9. I needed the money for a couple of important things. Like paying the rent and buying groceries.
Numbers eight and nine contain sentence fragments. In standard English they are perfectly acceptable. The fragment isolates and emphasizes an idea. In more formal English, they should be part of the original sentence or made into a separate sentence.
(I needed the money for a couple of important things, like paying the rent and buying groceries.)
10. We're going to have a big bash, and you're invited.
" Bash" is too slangy for a formal context.
11. The Smiths request the pleasure of your company at a banquet to be held in the honor of their son Josiah, who will be graduating from Andover College.
This sentence would be acceptable in a formal situation, such as a formal invitation.