Is "utmost" an adjective? I have the utmost respect for you?

Yes, in the sentence "I have the utmost respect for you." "utmost" is an adjective. It's actually a superlative adjective, like "best", so it's not preceded normally by the indefinite article "a/an". The word "utmost" can also function as a noun, as in "I've done my utmost to help you".