Example 1
Is there a place [for him to stay at]?
Example 2
This is an ideal restaurant [for couples to have a date at].
Example 3
He is going to choose a large venue [to launch the event at].
Example 4
This yellow blanket is her favorite blanket [to cozy in for the night].
Question 1
Are the parts in the brackets adjectives that modify the nouns before them?
Question 2
Do we need the prepositions?
No, they are not adjectives. An adjective is almost always a single word.
Many people would omit those prepositions, especially in casual speech. However, I recommend keeping them. I'll address each sentence:
If you omit the preposition ("Is there a place for him to stay?"), then it seems as if "a place" is the object of "to stay", which makes no sense. It is the same structure as "Is there a movie for him to watch?" (That sentence is reasonable because one can watch a movie; I don't know how one can stay a place.)
If you omit the preposition ("This is an ideal restaurant for couples to have a date."), then we don't have the previous issue because "have" already has an object. However, some people would object; the reason is related to why we can drop prepositions only in certain relative clauses. There is a discussion of that issue here: https://english.stackexchange.com/questions/565758/is-where-the-only-relative-pronoun-that-cannot-be-omitted-from-an-adjective-cl/565795.
If you omit the preposition ("He is going to choose a large venue to launch the event."), then it seems as if "a large venue" is the agent of "to launch". In other words, it appears as if the venue will launch the event.
If you omit the preposition ("This yellow blanket is her favorite blanket to cozy for the night."), then it appears as if "blanket" is the object of "to cozy". (By the way, I'm not familiar with the verb "to cozy" except in "to cozy up [to someone]". However, I assume that it isn't transitive.)
