It's really hard to say without being there in person. The wall certainly looks unstable and ready to be taken down, but a lot of concrete walls can look terrible, but still be structurally sound. We owned a house with a concrete block wall reinforced with rebar that had perhaps a 10 degree lean outwards that looked like it'd fall any day, but it took a LOT of sledge hammering, chiseling, jacking, and gas-powered diamond-blade cutting to actually get it out. It likely would have stood for another 50 years on its own.
In your situation, it appears that you've lost all the earth around your footing. At this point, it may be a lost cause and you'd be best taking it out. Ideally, you'd then replace it with a stepped retaining wall using a dry-stacked block. The dry-stacked blocks are a) easier to install and b) when installed properly will allow water to drain through rather than build up behind the wall pushing it out.
All that said, it appears that your neighbor has build a foundation right up along your wall. I don't know where you live, but I'd think in a lot of places that'd be against code (you typically can't build that close to the property line). In addition, it seems risky for them to depend on you to hold up their foundation with your wall. Seems that they should be bearing part of the ownership of whatever is retaining the earth on this property line.
Finally, if that's a down-spout in the lower photo emptying between the two walls, then a) take it up with the city, as that is likely also breaking all sorts of code regulations and b) your neighbor really should be worried about that. You don't want a downspout draining near the footer of your new kitchen's foundation.
If, when pouring the concrete right up against the asphalt, you end up with parts of the concrete that taper or feather out to less than say about 1.5 -> 2.0 inches thick you will have trouble keeping the concrete in one piece. The thin parts are likely to fracture and break off.
I suspect that if you rented a concrete type saw and cut off the asphalt in a nice square edge that you could then pour the concrete right up to it without a form. You will want to make sure the concrete is at least 3.5 -> 4.0 inches thick to make sure it stands up to be driven upon and changes of weather through winter seasons (if you live in a place where it freezes in winter).
Best Answer
No. And neither would the stuff I normally use; Fiber Expansion Joint, 1/2" x 3".
Tapcon a screed guide board to A (and/or B), at a height that will allow a board that's screwed to the top of the actual scredding board, to ride on it. That's how you screed something with only access to one side. And it's what you put the plank on to reach the corner to trowel it.
Ideally the guide board is an inch or so above the finished surface, otherwise you can't get a trowel in there. E.g., use a 2x4 for the guide and a 2x6 for the screed. Screw a scrap 2x4 to the top of the 2x6 so that it sticks out 1.5". Have fun jiggling it back and forth 3/4" at a time.
Frame as usual with 2x, well buttressed. Remove later, add expansion joint, and then fill/sculpt by hand.
You're SoL. The expansion material goes in before you pour. You cut them to length, put them in, and they fall over. Start filling and then stand them back up.