Well, in answer to your second question (why do internal combustion engines use combustion). The answer is pretty simple. If internal combustion engines didn't use combustion, they wouldn't be internal combustion engines. ;)
Your first question (Why do internal combustion engines use adiabatic expansion), I think there are a couple components to the answer.
- Adiabatic process are what you get when you do something really fast. And we want engines to do stuff fast. Translation: It's hard to let a piston get pushed up by exploding gasses without the expansion being adiabatic.
- The most efficient engine you can get uses the Carnot cycle which is: Isothermal heating, Adiabatic expansion, Isothermal cooling, Adiabatic compression.
Translation: Internal combustion engines use adiabatic expansion because the most efficient engine we can get uses Adiabatic expansion.
- As a quick aside, it might be worth asking, "If the Carnot cycle is the most efficient cycle, why don't we use that in internal combustion engines?" And there's a boring answer and an interesting answer.
The boring answer is: "Everything in a Carnot cycle must be reversible and we can't do reversible processes in the real world
The interesting answer is: "Because a Carnot cycle uses isotherms and those don't work well in an engine that can't just be submerged in a heat reservoir." Like you mentioned in the "extra info" on your question, combustion engines use isobaric or isovolumetric stuff. The otto cycle is: Isovolumetric heating, Adiabatic expansion, Isovolumetric cooling, Adiabatic compression (with a couple of real world steps like isobarically expelling exhaust and letting in more fuel and oxygen).
Why do we use those processes? Again, the simple answer is "because fast". When you light something explosive on fire in a closed container, the pressure rises so fast that the process is pretty much isovolumetric. When your piston is fully "expanded", and it opens up to outside the system to let in more air, that process is pretty much isobaric.