Are the Ten Commandments in OT arranged in the order of their importance to the believer? At Luke 18:18-20 we see:


  A certain ruler asked him, “Good teacher, what must I do to inherit eternal life?” “Why do you call me good?” Jesus answered. “No one is good—except God alone.  You know the commandments: ‘You shall not commit adultery, you shall not murder, you shall not steal, you shall not give false testimony, honor your father and mother.’


We see that Jesus is not quoting the commandments in the order of sequence they were given to Israel through Moses.

My question is: Are the Ten Commandments in OT arranged in the order of their importance to the believer, that is, if a person takes God's name in vain  and then commits adultery, will his former sin be considered  more serious than the latter one?

What do the teachings of Catholic Church tell us on the subject?
There is no biblical basis for the claim that the Ten Commandments are arranged in order of importance, and to my knowledge, there is no Roman Catholic teaching on the subject either. There are a couple places where Jesus does provide us with some insight into the degree of importance of the various commandments, but one stands out. Matthew 22:37.

It is important to notice that Jesus is not citing the Ten Commandments when he is discussing the two greatest commandments. The two greatest commandments are:


Love the Lord with all your heart, soul, mind and strength
Love your neighbor as yourself.


The first two commandments of the Ten Commandments are


No other gods before me
No graven images. 


Jesus considers the two "Greatest" commandments to be a summary of all of the commandments. 

If anything, the Luke 18:18 passage suggests that the Ten Commandments are not all in order of importance, since he freely rearranges them however he wishes.