Sorry if obvious.

I was wondering why he would take the wallet. While it may have been to use as proof he was the attacker, and then deciding to use the "rag" he improvised from the back of Stine’s shirt afterwards, one possibility is that he told Stine to stop the cab, and actually handed him a bill to pay.

One thing the attacker would have likely wanted to avoid is witnesses on the street. To lower the chance of any, he would want to slow down the driver as he is making his way along the street. He may well have given him a specific address, but more useful would have been a generic idea of where "I think it’s by Washington and Maple Street, I’ll recognize the house ok?". They head over to the general area.

When the cab is driving along the street, the attacker has a justification to ask him not to drive too quick, so he can look for the house, which gives him time to constantly look around without looking suspicious, he’s "looking for the house". In reality, he’s checking for witnesses.

Finally, he exclaims to Stine that he spotted the home, asks him to stop. This explains the odd angle at which the cab is parked, at a bit of a diagonal; Stine was put in a situation where he went from slower than cruising speed but not coming to a stop, to being asked to make a full stop right there, over a very short amount of time. Stine instinctively turns the car a little, as a cab driver would to not want to block the street and have their passenger come out in the middle of it either, it’s courtesy.

Chances are the attacker, to look less suspicious as he had asked Stine to slow down and just appear casual, already had his wallet and a bill ready. Why? Because if they stop, it gives him a bit more time as Stine looks for change. Now that they have stopped, the attacker does precisely that; he hands him a bill, and Stine gets his wallet out. That extra time gives the attacker just a bit more time to look around, and also lowers Stine’s attention to himself. His billet by now is in the wallet. He shoots Stine. He takes the wallet back.

The keys? He turns off the ignition, just to not leave a cab running, to draw a bit less attention.

It fits with the idea of him trying to leave no finger prints in place, which by all accounts he really wanted to avoid.