The proposed crossover next to the bus bay should likely be doable as it is only a residential driveway. Commercial driveways or driveways generating large number of turning movements aren't allowed to enter into bus lanes although yours would clearly be residential. However, it looks like a bus bay, not a bus lane to me (which further complicates it). I've never seen a driveway in a bus bay before and your local Council probably won't be keen for it although if you use good consultants they might be able to get it done.
Two major points I see Council having issues with:
1) How wide is that footpath adjacent to the bus bay? It looks less than 2.5m wide to me. If it is < 2.5m you would have to bring/cut the fence/gate back inside your property around the crossover location so that there is sufficient sight distance.
2) If you wanted to keep the fence/gate I imagine Council would require you to set the gate back say 5m inside your own property so that a car could park entirely in your property while you got out to open up the gate without a) obstructing the footpath and b) having the rear end of the car hanging out into the bus bay and obstructing it.
As Westminster mentioned above, some Councils will also have an issue of you reversing onto a road (i.e. you have to be able to exit front first), especially given that it appears to be a collector road. However, as it is a single residential driveway you might get away with it though, it just depends on the Council.
With the second proposed crossover in Frederic St I don't think you have much chance of getting approval. The first thing I notice is that on your plan you have located it where there is a power pole. You generally need at least 1m clearance from power poles (although your Council might vary a bit). You have no chance of getting a crossover approved to the right of that power pole (near the int.). On the left hand side of the power pole I can't see it being approved either. You generally need to be 6m back from the intersection kerb tangent point [i.e. where it starts to bend/curve] which puts you back where your existing crossover is basically.
