We've always been told that it is "preferable" not to go up and kiss the cross after the Divine Liturgy (nor venerate icons), IF you have partaken of Holy Communion.
Additionally, my mother always made sure we didn't eat anything that day that would force us to remove things from our mouths - such as fried chicken (on bone), ribs, apples (core), etc. Chewing gum was definitely a No-no!
Furthermore, my mom never let us take a "nap" when we got home, lest we slobber on our pillows!
I was always taught the "importance" of Holy Communion, and the respect It is due.
As for the cloth that is held beneath the Chalice, if some of the "Blood" has fallen on it accidentally (often happens with little children), the priest will usually "suck" it out afterwards. If that does not work, then he will burn it and use a new one the next time around.
One time some Holy Communion had fallen from the recipient's mouth, rolled off the cloth and landed on the carpeted floor. Immediately, once the Holy Particles had been picked up, that section of carpeted floor was marked off, so nobody would step on it. After Liturgy, that section of carpet was actually cut out, and burned, and the floor got patched with a bit of carpet cut out from the side of solea (where nobody would notice it missing.)
While God IS everywhere, His Gifts to us, must be treated with the utmost of respect.