Why were there three graves left behind at the cottage when Carol and Tyreese left (one with a small pair of shoes tied to it) ? They were alone and then baby Judith suddenly reappeared on the way to Terminus?

The grave with the little baby shoes on it was already there when they got to the grove. It explains why there was some baby stuff in the house (like the bed). And the two others are Mika's & Lizzie's.

When Carol & Tyreese left the grove, Judith was on Tyreese's back in a baby carrier. She was entirely covered, probably due to the fact that they didn't have the baby with them for the shooting that day. They often use baby boneka when they're doing shots, like when she's on Michonne's back while she's swinging her swords at walkers. Because they can't make bayi work too much and it's also easier for some of the shots.

Hope this helps!

A zombie wus one of them