market for lemons with signals - equilibriums vs constraints
The two questions you ask are related but distinct. First recall what a separating equilibrium is: its an equilibrium where distinct type takes distinct actions, supported by a system of consistent beliefs.
Now lets come to your questions one by one:

In the Lemons Market, a separating equilibrium would involve the seller setting different prices for the lemons and peaches. Can such an equilibrium exist? Think about what the buyer's beliefs would be and if the lemon can mimic the peaches?

I guess you mean Incentive Compatibility constraints instead of Participation Constraints. You must have encountered them in Principal Agent models. In these models, the agents play a (bayesian) game and the Principal can commit to design the rules of the game. The Participation Constraint ensure that the agents are willing to take part in the game designed by the Principal. The Incentive Compatibility Constraint ensures that the ensuing equilibrium of the game is a separating equilibrium - i.e. different types of agent will be choosing different actions in the game. The specific rules of the game designed by the Principal depends on the context.