Yes, unless your AP and ad hoc networks are on the same channel, you need two Wi-Fi radios to support this. AWDL "solves" this problem by quickly switching between these channels and schedules frame transmissions when the radio is tuned to the correct channel so the operation is (almost) transparent for the user (obviously, you loose some airtime for channel switching and your average latency increases).
Most cards nowadays support at least one ad-hoc and one infrastructure connection. Sometimes they have to be on the same frequency though, but it certainly wouldn't be hard for Apple to just buy one that supports multiple frequencies.
Not in most cases, but it is somewhat chip (and driver) dependent).
Miracast is another example of this - In most cases you can remain connected to your main network connection via Wi-Fi but also use Wi-Fi direct to run a casting session.
