The Russian participle "пробуждающийся" is formed from the verb "пробудиться". For instance, we can use this participle talking about nature in spring: "awakening nature"-пробуждающаяся природа. The synonym of the verb "пробудиться" is the verb "проснуться" and it's much more common. So, if you want to say, "I just woke up", you should just say, "Я только что проснулся."

you can say просыпаюсь if you're sleepy

In Russian, "пробуждающийся" has a poetic sense. You can say for example: "Природа пробудилась после зимней спячки". Talk about yourself better so: "Я только что проснулся", or " Я только встал", or if you just woke up, but feel that it would be nice to sleep a little bit more, you can say: "Я не проснулся еще".

1. Participles aren't used extensively in Russian. And English "is/am/are + ....ing" is always translated with the verb in the imperfective aspect form. So we have "пробуждаюсь". 2. Пробуждаюсь is ok, but if you don't want to sound too 'poetic', you should use more neutral "просыпаюсь"

No, you can't. You can answer Я только что встал and it will sound good. Пробуждающийся is used when you talk about something ( not yourself) that started waking up.