I think it's explained fairly well here. In this case, using the perfective implies that you wish to take a single action, and that you want to complete it (take a rest). Using imperfective here would mean that you just want to relax in general and it is not implied that there will be an end to this action.
It seems idiomatic to me. That's just the way you do it in Russian. In English, "I am tired" is present tense, with "tired" a predicate adjective, not a verb. "I got/became tired" is the parallel: past tense verb with predicate adjective "tired".
"I am tiring" is a present tense compound verb meaning "I tire".
Several on-line translators returned the same я устал for "I am tired", "I got tired" "I am tiring", and "I tire".
When I entered the present tense imperfective я устаю. I got "I am tired" "I tire", and "I weary".
So, it seems like in some instances you could use either past or present, but the idiom is to use past even when you're expressing a present state of being tired.
"I weary" on the other hand, is closer to "I am getting/becoming tired" or simply "I tire", so the Russian past tense wouldn't seem applicable for those expressions.
It is present in English. You're right that 'tired' is a past participle, but the verb phrase is 'am tired', which is the present perfect tense. Usually the present perfect is formed with 'to have' + past part., but here it uses 'to be'. This is a leftover from when the standard auxiliary for the pres. perf. could be either 'have' or 'be' - 'be' was used when describing a state or motion (similar to modern Dutch). If you read an English novel from the 19th century, you'll find verb forms like 'I am gone' and 'I am come' instead of the modern 'I have gone' and 'I have come'.
The simple past of 'to tire' is 'I tired'.
The verb for "nap" specifically is дремать, though it can also mean just to doze or snooze a little bit as well. Отдыхать is a more general "relax" verb - it can be napping, lying down, reading a book, playing games, watching TV, etc. In Russian there's the concepts of active (активный) and passive (пассивный) отдых. Active is typically something physical like going for a hike in the woods or managing your garden at the dacha, while passive would be watching TV or reading a book.
by the time I have read through all of the unnecessary gobbldigook in this thread, I now forget what it was that I wanted to see if it was discussed. so now, besides writing down the russian v's the english, I guess I will need a bl**dy note pad next to me to write down what I am curious about in the comments .... or better still, I think I will systematically delete my comments in all threads so as not to get notifications about new (usually gobbldigook) comments and here change the following to unfollowing the conversation once I have posted this. What a waste of my learning time grrrr