"There are a lot of boys."
"ii" should not be its own tile. It would make more sense to either have two "i" tiles, or one "i" and one "imasu."
This is ridiculous, why would you split a word on different tiles?
None of those answers are correct. You should not conjugate 多い to 多く unless you are negating it. Additionally, you would never use の to modify 多く、 as it has already been modified.
Both of the sentences I wrote above are grammatically correct. I know you just use adjectives that end with い before noun such as 大きい犬(big dog). But 多い(ooi) and 少ない(sukunai) are the exceptions and you often have to change them to 多くの(ookuno) and 少しの(sukoshino) before noun. Of course people can understand what you mean by saying 多い男の子. It's just a small mistake. There are some noun that you can use 多い instead of 多くの such as 多い給料 but I can't think of many.
I think @kaylovessasuke is correct.
おおぜい (大勢) looks like an い-adjective here, but actually it functions as an adverb (or noun?) here. You have to turn 多い into the adverbial form 多く for use with います (to exist).
Be careful when using 大勢「おおぜい」since it is used to refer to a crowd or large number of people as opposed to objects.
I totally agree with you. It's misleading to divide words to different tiles.