The renovation lefthas only one task left, which is caulking.

I got a message that the worker will/would arrive at my house around noon.

To be super correct grammatically, we would pair "would" with the past tense.

I headed home to let heim in during my lunchtime.

He is receiving the action like a direct object (let him in, give him keys, etc.)

Today wase had heavy rain.¶
Today it was raining heavily.

"is rain" and "was rain" are incorrect. "is raining" and "was raining" are correct

I didn’t have an umbrella, but I wore thea raincoat.

"the" implies you have already told us about this raincoat, or we already know about it. These kinds of distinctions can be very hard for non native speakers, but I think it is generally true that we use "a" or "an" to describe clothes and accessories people are wearing.

I walked in the rain and, of course, I allgot wet.

We use the expression "to get wet." In the US, a common expression is "I got soaking wet" or "I got soaked."

I cleaned the countertops to make easy for caulking while I amwas waiting him.

He picked out some tools from tool backg and started caulking.

"pick" alone is understandable, but it tends to be used most often for picking fruit from bushes or trees or for picking flowers.
However, you can pick a number in a game, pick a color, etc., so it is not a very clear distinction.
"pick out" is good to use for choosing some items from a collection. But if you have doubts, you could use "he chose" or "he selected"

A wWhite color onecaulk was put along edges and sides, and a clear one was put oncaulk around the sink.¶
He used white caulk along edges and sides and clear caulk around the sink.

I think it's helpful to remind the reader: using the word "caulk" keeps us from being momentarily confused.
English prefers the active voice. The passive voice should only be used purposefully for effect.

After that, he repaired a wall. with some pink muddy item(sticky liquid?)

My general rule of thumb is: if it's not essential, it's ok to leave out.
(I have no idea what you might be referring to)

They congratulated me [so much]!

This just doesn't sound like what a native speaker would say. Unfortunately I can't think of a good expression using "congratulate" or "congratulations."
I might say "They were so happy for me" "They thought it looked awesome." "They were amazed/very impressed"