1) You only need to add "me" if there's some ambiguity as to whose hand is going up. Under normal circumstances, you're raising your own hand, and no one's arm is intertwined with yours creating confusion. Thus we say, in a big committee meeting,
Levanto la mano [para pedir la palabra] | I raise my hand [to get a turn to speak]
However, if it's a pronominal verb, e.g. lavarse las manos, then it would be
Me lavo las manos.
Maybe you have gotten the pronominal pattern stuck in your ear. If so, it would be helpful to look at some additional non-pronominal body-part usages, to establish that second pattern. Here are some: Cierra los ojos y pide un deseo; puedes abrir los ojos ahora; no es cortés poner el codo en la mesa.
It's also possible you are confusing with some other phrases involving mano that you heard. Perhaps one of these? Dame la mano | Give me your hand; No me levantes la mano | Don't raise your hand against me.
2) You will need an indirect object to say "Open his eyes". Example in a medical context:
Ábrele los ojos [al paciente].
3) "Me veo en tus ojos" is fine as a description of a poetic image. (It can't be literal, right? Because someone's eyes don't actually function as mirrors.)
Your proposed "Me te veo los ojos" doesn't sound right to my ear. Here's an intuitive attempt to explain it: Here, both object pronouns would be indirect; but there can only be one indirect object pronoun in a cluster, because if there were two, the listener wouldn't know which pronoun to connect with which object. In this case, for example, how would the listener know whether the eyes refer to "me" or "te"?
The other objection I have to your proposed sentence is that the "en" is gone. Your sentence no longer conveys the poetical image of the other person's eyes functioning as a quasi-mirror. It's just an objective description of seeing the person's eyes. The poetry is gone.
(a) Ábrete los ojos tú mismo. (I'm assuming you want to convey that the person should open his eyes without someone else helping him. Correct me if I misunderstood.)
(b) Almost. You've made a little mistake with the gender of the article. Remember that manos is feminine (irregularly).
If you fix that article, I would not be very uncomfortable with your proposed sentence; however, I would prefer a tiny change in the preposition:
Él le lava las manos por ella.
Por/para explanations have been treated elsewhere and I won't get into a treatise about that here. However, if the existing Q-As on that topic leave you with some lack of clarity, that would be a great separate question to ask.