I am often confused about the construction of clauses with to.
I saw that there was not much water left in the bottle, so I picked it up went to the kitchen refilled it.
Is this a possible contraction of the previous sentence? Is it acceptable?
I picked up the bottle to refill it.
Similarly, is the following sentence acceptable?
I saw the show to talk about it next morning at work.
Your first sentence requires at least one comma and an and:
I picked it up, went to the kitchen and refilled it.
"So* looks back to the previous clause to supply a reason—because the bottle was almost empty—for doing all three things which follow, but does not explicitly state that you picked it up in order to refill it.
Consequently, it cannot be regarded as a ‘contraction’ (I think what you mean here is an ellipsis): it is a restatement, which asserts a purpose leaves out its realization in the two following actions: taking the bottle to the kitchen and actually refilling it. The sequence might have been interrupted before you completed it.
Likewise, the second sentence, I saw the show to talk about it next morning at work, asserts only your viewing and your purpose; it does not assert that you did in fact talk about the show the next morning.
I saw that there was not much water left in the bottle, so I picked it up went to the kitchen refilled it.
There is a problem with this sentence: you need to add commas.
I saw that there was not much water left in the bottle, so I picked it up, [I] went to the kitchen, [and] [I] refilled it.
In very casual, informal writing you might leave out and, but it is normally compulsory. Leaving out I in the second and third clauses is perfectly normal and not informal at all. There is nothing unusual about to: it just introduces the direction of went, an adverbial phrase of direction.
I picked up the bottle [in order] to refill it.
This is perfectly normal. You don't need in order: it means exactly the same thing. In a simple, clear sentence like this, using to to introduce a purpose with to + infinitive is standard. It has nothing directly to do with to as in the first sentence, where it introduces a direction.
I saw the show to talk about it next morning at work.
This sentence is of the same type as the previous one: the reason/purpose why you saw the show is that you could talk about it the next morning. The situation is a bit unusual (is that really the main reason why you saw the show? Isn't that a bit silly?), but the grammar is fine.
