The Polish ch sound (or h) is similar to the Spanish jota (j in trabajar) or the German ch in machen. It's also similar to the English h, but more pronounced, and comes more from the throat
If you mean can 'ich' mean his or her singular as it often does in English, then no, I don't think so. It is only used to mean 'belonging to them' (plural), but the 'them' could be masculine, feminine, neuter or a combination. Please somebody correct me if I'm wrong.
If ich/je can mean "their" as in "their dog," does it make grammatical sense to say "ten pies jest ich" to mean this dog is theirs?
Yes, that's correct.
Wait... only "ich" means "their". "ich/je" are Accusative forms of "them". The fact that one of them is also a possessive is better to be treated as a coincidence.