For example, take these two sentences:
"I wasn't impressed with it."
"I wasn't impressed by it."
Which would be more formal/proper?
"I wasn't too impressed"
or
"I wasn't that impressed."
Which would be more formal/proper?
The more natural sounding one or at least, the one I hear the most is "I wasn't impressed by it". "Impressed with it" allows for an odd structure which gives way to another definition for impression, to seal; to imprint.
As for your second question, a lot of people use either or expressions very loosely. To be "impressed" is to evoke admiration out of others.
I somewhat oppose certain adverbs modifying verbs/adjectives such as "very bad", "really good". Some of these words do quite well on their own. Bad is bad, there isn't any "less bad" nor is there a "less good" concept, I think.
Regarding your word however, one can be slightly impressed.
"I wasn't too impressed" can be used as a declarative statement.
"I wasn't that impressed" can be used to enforce your statement, making it absolutely clear- Hey, I think it was entertaining but I wasn't that impressed by it.
Both definitely sound fine to me as a native speaker, and I'd go so far as to say that they are both correct. However, I think there is a slight difference in meaning - "I wasn't impressed by it" looks rather a lot like our (English's) passive construction, and so the preposition "by" really underscores that the thing itself failed to impressed you. "Impressed with it" means more that you evaluated the thing and found it unimpressive. Most people, however, would be hard-pressed to decide which of these two options more suits what they would like to say, and so it is safe to use them interchangeably.
Any of the options you mentioned are correct, but if you specifically want to sound formal or proper, you should avoid using contractions, and say "I was not impressed by it/with it" or "I was not very impressed."
