Not sure on the вокруг vs кругом

Увидеть because the whole event is wrapped up, seeing happens and it's over. Sort of like: he is not hoping to be seeing him, he's hoping TO SEE him, as a solitary whole event. Imperfective and perfective are super abstract, admittedly, though are not without reasons.

его is used because Russian allows the Genitive case special power when talking about absences and nonexistence. Often can be found cases where what you would think would be the subject, is put in the genitive, and the verb is conjugated as "impersonal", which just looks like third person singular. Certain verbs do this when the person in question isn't logically actually DOING anything, emphasizing that it's out of their control. A good example is спаться - to sleep.... Or.... To be slept... спать with cя added--

Ему не спалось - he didn't sleep, here it's an impersonal verb [whose subject you can think of as some sort of omitted "it" if it floats your boat], and the dative is used. These sort of constructions make more logical sense when you regard the cases in their raw conceptual sense. Dative is a bit more attributive.
It was not slept to him, sleep didn't come to him

Он не поспал - he didn't sleep - but here it's more as though he just didn't go to bed and stayed up

One of the Genitive's many traits deals with absence.

У меня есть карандаш
У меня нет карандаша [some teachers explain нет as sometimes being a combination of не есть ]

I have a pencil
I don't have a pencil

У меня был карандаш
У меня не было карандаша

I had a pencil
I didn't have a pencil

Usually it's a good idea to think of things in a sentence by sentence grammatical sense, when learning Russian. In this case however, try to think of things in abstract inherent meanings. Imagine you were born in this language where you have these cases, using it for years, and it follows that theses cases would start to form a small mind of their own.