As far as trowells are concerned, I'd usually recommend, whatever you're used to. My normal choice is a Northern pattern, (Like a London, just a bit broader), but I sometimes use a Candian pattern for thicker walls and on really thick walls, I get the labourer to dump the mortar on with a shovel, then I spread it with the trowell. Since you're not used to anything in particular, what you've got is probably as good as any.
That mortar does look a bit strong. Examine the mortar in the edge of the exposed broken brickwork. Does it look the same colour right through the thickness of the outer skin. If so, I'd go for a 1:4 mix (cement to Building Sand). If the main body of the wall is lighter in colour, I'd go for a 1:6 mix to lay the bricks, rake out the joints, then point up with a 1:3 mix.
Since you have the obstacle of the soil pipe, I'd be inclined to break the bond and go for quater bond instead. To do this you would remove the bottom two bricks, the top two bricks, the half bricks on each side of the opening, and then, with an angle grinder, cut the other four bricks back to 3/4 length (looks like the silicone is marking where to cut).
Your new brickwork would then be alternating courses of 2 full bricks followed by 3 half bats. (See my crude emulation below). When cutting the 3/4 length bricks, note that the gap you need for three half bats and their joints will be about 350mm.
When cutting the half bats, just cut the bricks across the middle, then lay with the cut edge into the wall (ie. the original end of the brick will be exposed on the face of the wall). Make sure that you have a clear gap between the outer and inner skin. If the half bats come out too close for comfort, chop a bit more off them. Don't worry if they aren't quite the full thickness of the outer skin, you can easily get away with them being 30mm too short, without any worries.
Once you lay the bricks, don't attempt to cut off the mortar that squidges out on the cavity side. Like as not, you'll drop it down the cavity (where it can pile up and cause a breach just above the DPC). I know it looks like a runny nose just waiting for a tissue, but if you Just leave it hanging, it will do no harm.
When laying the joint for the top course, don't overdo it. When laying the a brick for the top course, fill the frog and put another joint on top of it. Make this joint just about 20mm tall and only along the cavity side edge. As you push it into place, it will naturally get spread out across the top of the brick. Put your trowell against the top of the brick to catch the excess that gets pushed off of it (This will probably be most of it
). To press it down into position you may find it handy to poke a cold chissel through the top joint to act as a lever. Just accept that you will have to fill the top joint by forcing the mortar in afterwards.
I've just noticed that the mortar on the inner skin looks the same as the outside. Chances are it's about a 1:4 mix. Don't forget the plasticiser. The fairy liquid bodge is no good unless you are using a mixer (I take it you're not).
The jointing is run of the mill rubbed joints. Find any plastic or metal item with about the right cross section to rub them with. I find the oval cross section of those stanley screwdrivers good for small areas. For thin joints you use it with the narrow edge into the joint, For thicker joints you use the broader side. In between sized joints, use it a somewher inbetween. (No good for large areas as they quickly become rather scratched). By rubbing narrow joints deeper, it disguises the difference in joint thicknesses.
If you take more than, say, an hour to lay the bricks, rub the joints that you've done so far (vertical joints followed by horizontals). About 20 minutes after the rubbing, brush with a broom, (using horizontal strokes).
Just one minor point on the inner skin, instead of rubbing the joints, leave them raked out (about half an inch should do). This will act as a key for your render/plaster.
Structural supports? No I don't think so. You could, of course, save all this trouble by just fitting a cat flap.