The only context these might differ is in a more complex sentence, such as one in which "where" might not refer to a physical place. Otherwise, you are correct, these have the exact same meaning, and do in the sentences you have provided.
As a possible example of an exception:
I wrote an article about playing basketball where I discussed the potential health benefits of the sport.
Note this would have substantially different meaning than:
I wrote an article about playing basketball at a place where one could remain outside of direct sunlight.
In the first, where refers to the article written. In the second, where refers to the place one is playing the sport. In the second case, at a place helps clarify the meaning.
Yes, they are both grammatical, and have very similar meanings, even though their structures are radically different. (Two minor corrections, which don't affect what you are asking about: "walked in his direction", and the idiom is for dear life. Alternatively, you could say "for his life")
In "On seeing that ... , he turned", the initial clause is a prepositional phrase locating the action in time. It is functionally equivalent to "After lunch" or "at three o'clock". It modifies the whole predicate.
In the second sentence, "Seeing that ..., he turned", the initial clause is either an adjectival clause modifying "he", or possibly an absolute clause modifying the whole sentence. But it may be doing more than specifying the time - it might, for example, imply causation or motivation.
The other three sentences are a bit more complicated. I find them all a bit unnatural. In the first one, "He got up from his seat, and walked towards her, seeing that she had arrived", I find it so unnatural in this sense that for me a different, idiomatic, use of 'seeing that' appears: this means 'even though', or 'because'. So the primary reading of that to me is something like "Because she had arrived, he got up ...", with an implication that he didn't want to, or wasn't ready to, but felt obliged to because she had made the effort, or something like that.
The last two are fine, but as I say they feel a bit unnatural to me. I find the last one, with the comma, slightly better.
Best Answer
Yes, they are both grammatical, and as far as I know they have the same meaning.
I think "Check this place out" is more common for prosodic reasons: it puts a strongly stressed word at the end. "Check out this place" puts emphasis on "this", and suggests "as opposed to [some other] place".