かける can be used as an subsidiary verb to mean "start to [verb]", so 崩れかけた is indeed the 連用形 ren'yōkei (masu-stem) of 崩れる followed by かけた.
崩れかけた門 means "a gate, which has started to break down / deteriorate". Of course you would usually translate this more directly, maybe (for lack of a better word) "deteriorating / wrecked gate"....
As a non-native English speaker myself, I advise you not to try to understand the English translation, specially if English is not your native language. I think it's more useful to just understand the situation in which you can use each expression, so you don't need to rely on other languages to understand those expressions.
Let's imagine you are appointed ...
In the Japanese language, there isn't the difference between the "make" and "let" usage of causative verbs. We only judge it depending on the context.
For, example, 私は、無理やり子供に野菜をたべさせた is translated as "I made my child eat vegetables (forcibly).", 私は、自由に子供にお菓子を食べさせた is translated as "I let my child eat sweets (freely)."
As A.Ellet pointed out in a comment, 「私は僕が」 doesn't make sense, so I modified some of the pronouns (or replaced them by nouns) in your question to make the sentences valid. Also, note that I placed pronouns that would be usually omitted in a natural conversation between brackets [ ] . Finally, I also added some complements 野菜【やさい】/ アイスクリーム to make the ...
Generally よ is a kind of intensifiers so "a infinitive verb + よ" will strengthen the meaning of the verb.
In your example なる (なり isn't an infinitive though) means "to become", so なるよ! will be "surely become ... !"
Looking just at the first part:
Let's break it down word by word and then reconstitute it.
Verb: 着【つ】く (tsuku, "to arrive") in the conjunctive ～て conjugation.
Particle: "because; from". Following a verb in the conjunctive ～て conjugation, this means "after [VERB]ing,..."