How can I test if a table is opened exclusive or shared?

The easiest way is to wrap the USE command in a TRY/CATCH because the table might be open with a different alias.

How can I change it from exclusive to shared?

Close the table and re-open it.

If I execute a USE mytable SHARED command, I will cancel a previous SET ORDER TO command, which I do not want to do.

Just set the order again after you re-open it.