I'd say initially it would go very much like in StarCraft but probably worse for the Teminus system. The Terminus systems are known to not like The Citadel and aren't under their jurisdiction. Because of this I'd say The Citadel wouldn't give two shits about what happens to that area. So lets just fast forward. The Zerg have a couple of hive worlds in the Teminus and are starting raids on Citadel Space.
The Overmind quickly realizes that ships in ME rely on the Mass Relays to quickly travel between systems so they send strike forces to take out the relays before the invasion actually arrives. A few worlds fall to this tactic before The Citadel realizes this and starts having armed guards around the relays...By now the zerg hive worlds are pumping out zerg by the truckload and there is no real way for the ME races to actually stop them because they can't travel there. This means that eventually Citadel forces start getting stretched thin by all the zerg invaders. Between protecting mass relays, protecting the planets and scouting for new invasion points ME just doesn't have the resources to defeat against the zerg...and they can't really attack. This means that the zerg will inevitably take the entire galaxy...the only question is who the hardest race to destroy is. I'm gonna say its the Salarians because they are known for Biological warfare and by now probably have enough zerg specimens to develop something like a virus against the zerg. But overall 10/10 for the zerg.
R2: The battle would come down to how well ME biotics users can fight against her, or at least keep her occupied...My guess is not well enough. Kerrigan is the most powerful psy user in the SC universe, She was depicted as being able to take out 27 tanks solo and she casually crushes battlecruisers in SC2. If by some miracle Jack, Smara, Liara, biotic Shepard and Miranda can keep her occupied the others could MAYBE kill her....Kerrigan 9/10
Round 3: SO I'm gonna go on the idea that ME just sends EVERYTHING before the Mass Relays get blown up by the zerg. This way I would actually say ME wins. Krogan soldiers are a scary bunch and I'd say a squad of them can easily take out an ultralisk. Turian, Human and Batarian fleets can maintain air control and during all that time Salarians develop their zerg virus. I'd say ME wins 8/10