El ver la lluvia desde mi cuarto me relaja.

Because Spanish normally uses the infinitive when an action is the subject of a sentence. In this sentence, seeing the rain from my room is the subject, so ver is the normal choice.

Viendo la lluvia usually works more like while watching the rain or watching the rain, not as the main subject in this kind of structure.

Because I am not the subject of the verb relaja. The subject is the whole phrase El ver la lluvia desde mi cuarto.

So the structure is:

El ver la lluvia desde mi cuarto = the thing doing the action
me = the person affected
relaja = relaxes

So it means Seeing the rain from my room relaxes me.

No. Here me is an object pronoun, not a reflexive one.

• Me relaja = It relaxes me
• Me relajo = I relax / I am relaxing

So in your sentence, me means me, the receiver of the action.

Spanish often uses the definite article with things like weather, natural phenomena, or things seen as a whole. So ver la lluvia sounds natural as see/watch the rain.

Saying ver lluvia is not impossible, but here it sounds less natural and less complete.

It means from my room and tells you the viewpoint of the action ver la lluvia. In other words, the speaker is in their room, and from there they see the rain.

It describes where the seeing happens from, not where the rain comes from.

Because desde is the normal word for from when you mean a place or viewpoint.

• desde mi cuarto = from my room
• de mi cuarto would not naturally express that same idea here

So desde is the correct choice for the location you are looking from.

Yes. In everyday Spanish, especially in Spain, cuarto often means room, and mi cuarto commonly means my room or my bedroom.

It can also mean fourth in other contexts, but here the meaning is clearly room.

Yes, but the nuance changes a little.

• ver = to see
• mirar = to look at / to watch

So ver la lluvia is a bit more general, while mirar la lluvia suggests more deliberate attention. Both can make sense, but ver is what this sentence uses.

Because the whole phrase El ver la lluvia desde mi cuarto is treated as one single idea. Spanish sees that entire infinitive phrase as a singular subject.

So you get:

• El ver la lluvia desde mi cuarto me relaja
  not
• ... me relajan

Yes, absolutely. That version is very natural and common.

The difference is mainly emphasis:

• El ver la lluvia desde mi cuarto me relaja emphasizes the act of seeing the rain
• Me relaja ver la lluvia desde mi cuarto emphasizes the effect on me

Both mean essentially the same thing.