We use the apostrophe for where we drop letters, or to signify ownership, in other words what would be genitive in Greek. We generally don't contract "is" in English, apart from "it is/it's". "The piece's red" would only work if we used it in a genitive sense, and that would require another noun, as in "the piece's red colour".
All these sentences sound fine to me - but I may not be a representative English speaker! Would you avoid the contraction in all of these?
The car's in the garage
My mom's going to ground me
The new Greek tree's 90% complete
She's coming tomorrow
(I don't think I would write the piece's red by the way, but the reason for that is more a phonetic one - it feels odd to write "piece's" when "piece is" is identical in sound.)