You can use "put" and sometimes "set". It is not very common but it is far from uncommon. I'd say put is more common and set, in this context, is less common, so it is somewhere in between. Hope this helps :)

Set, put, lay, are synonymous with place. Also fix and emplace :)

"Place" is a common verb. It means the same as "to put in place". It can refer to physical or conceptual location. Here are some examples. "Let's place Sally next to Paul at the table so they can talk more easily" (physical placement) "Jack? The name sounds familiar but I can't place him." (means I can't remember how I know him or where I met him) "The runner placed third in the race" These are but a few examples. You can find many more - it is a useful and flexible verb.

It means to put something in a particular place as a verb, but it’s not very common. Put, set, or lay can also be used which has similar meaning.

It's not rare. It's not common. The obvious alternative is "to put." It would help if you told us the context. What are you trying to say? I have the impression that Spanish speakers are more concerned with avoiding repetition than English speakers are. You may not need to use a different word. "To put" is a simpler word and just indicates broadly where something is going to go. "To place" sometimes suggests a relative position among several possibilities. Or, it suggests a careful, gentle, and precise action. "Put your coat in the cost closet." "Honey, where did you put the bread? I can't find it." "I put it in the freezer." "In a symphony orchestra, the violins are placed on the left, the cellos are placed on the right, and the percussion is placed in the back." "When setting a table, place the fork to the left of the place, and the knife and the spoon to the right."