Learn Italian: Questo albergo è troppo caro. - This hotel is too expensive.

In this sentence, no. Here, troppo acts as an adverb meaning "too" (modifying the adjective caro). Adverbs never change their endings. However, if troppo was describing a noun (meaning "too much" or "too many"), it would change to match that noun. For example, "too many hotels" would be troppi alberghi.

I thought questo drops its last letter before a vowel. Why isn't it quest'albergo?

You can write it either way! Dropping the -o and adding an apostrophe (quest'albergo) is very common and natural, but writing the full word (questo albergo) is also completely correct grammar and often used in writing.

Does caro mean "expensive" or "dear"?

It means both! Just like in older English, caro can describe something that costs a lot of money, or someone you care about deeply (like starting a letter with "Dear..."). Since we are talking about a hotel, "expensive" makes the most sense here.

If I wanted to say "very expensive" instead of "too expensive", what word would I use?

You would use molto. The sentence would become Questo albergo è molto caro. Just like troppo in this sentence, molto is acting as an adverb here, so its ending wouldn't change.