Huh. I hadn't actually heard of "soak up" being an intensive form of soak, though I could understand it. Generally soak means "completely absorb and fill, especially through presence in something." The image in my head is of a sponge, which holds water when you soak it.
When I hear soak up, I think of this example:
He used a towel to soak up the spilled drink.
Here, it means to draw up inside. It also makes sense for someone to soak up something pleasurable, which means to take time to enjoy something, letting it distract you completely. This idea of words that mean filling equalling enjoyment is pretty common in English, as does equating emptiness with sadness or depression. (Like drained.)
I'd say that for most people, soak means completely, even if only in a hyperbolic sense. (Like, not actually soaked, but really wet, at least.) So that laptop in your last sentence is almost certainly ruined, like a full cup of water was spilled on it. "Soak up" is not at all necessary to express degree, just how you feel about it, and it doesn't always make sense where soak would.
As for your examples, the ones I don't mention are correct:
The rain soaked up my clothes.
I prefer "The rain soaked through my clothes." That means that the water got all the way through to your skin all over, and it's a clearer image than just "soaked up".
I got soaked in / with / by the rain.
In or by is fine, with is awkward when talking about water, because people assume it's water. If you got soaked with apple juice on the other hand, then you should say "with apple juice".
I got soaked up in /with / by the rain.
same note as above, I don't like "up" again because it's very generic, and soaked is fine if you put the right emphasis when you say it.
Use the paper towels to soak up the tea spilled on the table.
I corrected this, pour usually means intentionally, and make sure to use past participle about something that's already happened.
My laptop was soaked.
Unnecessary to say "with water" because people will often assume that, as I noted above.
And soaking up sun is perfectly fine, soaking up rain is not, unless you're standing in it directly, and enjoy that. Some people do, and would use that phrasing, but it's not what you mean. Soaking up the air after the rain is something I haven't heard before, but I like it.