Int. Jp 2-1 もう〜ている・てみせながら・目を覚ます

Hello everyone, I have a few questions from yesterday and today's studies.

1: Verifications of my Translations - Scene-setting pre-conversation descriptions
第一：「キャロル・ベーカー、朝キャンパスで日本語の三田先生に会う。」
'Carol Baker meets Japanese teacher Mita on campus in the morning.'
第二：「朝８時に目覚まし時計が鳴って、キャロルが寮の部屋で目を覚ますと、同室の日本人の留学生ゆりが話しかける。」
'The alarm clock rings at 8am, and when Carol wakes up in her dorm room, a Japanese exchange student in the same room named Yuri talks to her.'

2: もう〜ている
Does this construction mean 'Already did ~'? I believe it was in the context of 「今朝もう会っている」, 'We already met this morning'. If so, is it possibly to say things like, もう食べている (I already ate) or もう十分寝ている (I've already slept enough)

3: 「会釈をしてみせながら、」
Came across this ~te misenagara. Does it mean 'While showing ~'?

4: なかなか〜ない
Nakanaka has always tripped me up despite hearing it so much. I understand from my textbook it means 'no easily' or 'takes a while' when used with the negative. Is this usage correct: 「ゆうべ宿題が多いあって、なかなか出来なかった。それで遅れたんです。」'I had a lot of homework last night, and I couldn't do it easily. So that's why I was late.'

5: 「起きる」と「目を覚ます」
Difference between using these? Is one more casual than the other? Are there any rules to using me o samasu I should know? For example, can I use me o samasu to talk about MYSELF waking up?

よろしくお願いします。

1)
The meaning is not so different, but 朝８時に目覚まし時計が鳴って is also conditional.

「朝８時に目覚まし時計が鳴って、キャロルが寮の部屋で目を覚ます」と
When the alarm clock rings at 8am and Carol wakes up in her dorm room,
(= When Carol wakes up in her dorm room after the alarm clock rings at 8am, )

2)
Yes. ～ている form expresses experience in those cases.

3)
In both meanings, yes. ～ながら also can be used for adversative conjunction "although", as same as "while" in English.

会釈をしてみせながら、手を振った。(and)
会釈をしてみせながら（＝会釈をしているけれど）、心の中では憎んでいた。(although)

4)
ゆうべ（は）宿題が多くて、なかなか出来ませんでした/終わりませんでした。それで遅刻したんです。

Yes, for なかなか～ない.

5)
目を覚ます vs 起きる is close to "to wake up" vs "to get up", but they are partially overlapping. 起きる is also used for "to wake up". Plus, 目を覚ます also means "to realize/disabuse", and 起きる can mean "to get up from~", or "to be awake" in the form of 起きている. 目が覚める is also used for the speaker themselves.
e.g.
目を覚ませ。お前は騙されている。(to realize/open your eyes)
ソファーから起きて、台所へ向かった。(to get up from~)
一晩中起きていた。(to be awake)

I made sure to make notes from everything you've given me here.
Thank you for all your help, as always.