In most contexts, in the first place [assertion A] is a somewhat metaphoric usage where first doesn't really have anything to do with the temporal sense of A being in some way earliest in time.
It's usually used in contexts where there are several relevant factors that might be mentioned by the speaker. But assertion A is the most important one (and/or the one from which further supporting assertions logically follow, and the speaker intends to go on and make those further assertions).
In short, it's sense 2 in oxforddictionaries: first - Foremost in position, rank, or importance.
USAGE NOTE: The metaphorical usage referred to above almost always occurs before the relevant assertion. Consider...
1: In the first place John was there.
(The fact that John was there is central to the points I'm about to make.)
2: John was there in the first place.
(John was already there before whatever happened next that I'm going to tell you about.)
It's also worth noting that usage #1 generally occurs in contexts where the speaker is refuting, or disagreeing with someone else's prior assertion (you wouldn't be so likely to use it as the opening line to start a conversation or make a point).
The idiom to be cool with a thing means to not find the thing objectionable.
Example:
"Would you be cool with dropping by the post office on the way there?"
"Sure, no problem."
Another example:
"I thought you would be cool with inviting her. I didn't realize you had some feud going on."
Please be warned, that American slang usage of "cool", "hot", and "lukewarm" to indicate enthusiasm or agreeableness can be complicated and at times contradictory. To say that someone is "cool towards" someone else means they are expressing disdain; to say someone is "cool with" someone else means they don't find that person objectionable, and will get along collegially with them.
Best Answer
The common English phrase "I'm beginning to think that ..." is kind of a standard or set phrase, and the meaning does not normally change with context. It is not in any way a uniquely American expression, as it's used by all English speakers. It simply means:
or
This structure can be used with many cognitive verbs:
The difference between "I'm beginning to think that ..." and "I'm thinking that ..." is the nuance of recent or just started. It often implies a change in thought based on new information.