Are Roman Catholics required at any time to bow down to statues or images? Are Roman Catholics required at any time to bow down or make obeisance to or kiss statues, images, icons, etc. (i.e., inanimate objects)? 

If so, is it considered a sin not to?
During Mass, as the occasion warrants, Catholic priests/deacons (not the lay faithful) are required to: 
Bow before the tabernacle the first time they cross it.
Kiss and incense the altar and the Gospel book during mass.
Incense the Cross and the tabernacle
Every Catholic as a sign of belief and respect has to bow before the alter, genuflect before the reserved Blessed Sacrament and kneel if the Blessed Sacrament is exposed. This is done every time they cross these when moving from one place to another. Priests are excused from this norm when celebrating Mass.
In some Asian and African countries during mass, the sign of peace is exchanged by bowing to each other because of their culture. (In western countries they shake hands, I am not sure about the universal norm for this. GRIM 82 leaves it to the local Bishops' Conference.)


Ignoring these without a grave reason and on purpose is considered to be Sin (provided other conditions are met). Note that these are just customs, no one is going to accuse a lay person if they don't follow these things; In-fact if you don't do this because you don't believe in what church teaches, then you have greater problems than worrying about these actions. But if you do believe, then as Card. Arinze says,


  "If you believe that Christ is our God and He is present (in the Blessed Sacrament) why don't you kneel? Why don't you crawl? Why not show? Show respect."


Other than this there is no obligation on the part of Catholics to kiss or bow down to anything else (including persons).


It is to be noted that on Good Friday, during the veneration of Cross, the Cross is kissed. This is mandatory for priests. It is strongly recommended for lay faithful. But not obligatory.