Why are the já and ty required here?
When the subject changes within the sentence (or in the following sentence), we use the pronouns so that it is clear who we talk about.
The correction that I was given was "mám chléb a ty sůl." Is that really correct, i.e., can one omit the máš here?
Yes, I believe that "máš" can be omitted here and, funnily, I almost wrote what you got for your "correction." :-)
UPDATE 4 Jul 2018 ---------- I just used "já mám chléb a ty sůl" and it was accepted.