Is the meaning different if I say отдыхать instead of отдохнуть ?
I'm not sure, but I think that отдохнуть is perfective. So you would use it if you are tired and want to rest. But if you are e.g. planning to lie in the bed all saturday then you would use отдыхать.
Isn't this past tense?
Устать is generally used in the past tense. You could think of it as "to become tired". If you're tired now then you must have become tired at some point in the past, so it's correct to use the past tense.
Может я конечно и ошибаюсь, но по моему это стоит переводить как: "Я уставший ...". И тогда это будет настоящее время. Но так редко кто из русских говорит. Чаще говорят "Я устал"
This would be an example where you need to drop the pronoun for the second verb? Or should "Я устал и я хочу отдохнуть" also be accepted?
It's grammatically correct, but you already said я and it's clear from the context and verb form that you're still talking about yourself so it's more normal to drop the pronoun the second time.