что стоял около домика - it's just added information, you don't put any empasis on the clause that way.
который стоял около домика - it looks like important information, you put emphasis on the clause and you emphesize the fact that it was a table that was close to the house.
Anyway, it's kind of grey area. It's difficult to explain completely. It's more like a feeling.
Perhaps this would help. You know that there is a difference between these two English sentences?
The car that is red is very speedy
The car, which is red, is very speedy
Maybe it's not absolutely correct to say this but the difference between THAT and WHICH clauses are very similar to the difference between those Russian КОТОРЫЙ and ЧТО clauses
As for the визит word. Well, I don't know what to say. На визите just feels absolutely wrong, don't know why though.