Which country English-speaking Englishcountry should I say?

Not sure if what you had was wrong, but hyphenating the term and using it as an adjective seems more natural.

To be honest, I'm not good at the study and memorizesation of what countries there are in the world.

"memorisation" is the noun you need if you want to use this construction - but it would be more natural to use gerunds, "I'm not good at studying or remembering countries".

(And I'm not sure what I should call in English thatthose kinds of studies. in English.)

Or "that kind of study", match singular-plural that-those.

("Geography" :))

Thinking of "countries speaking English," that remindeds me of a board game called Scotland Yard.

Better to match tense here. Could be "When I was thinking of ... it reminded me" or "When I thought of ... it reminded me". I think "that" is okay but "it" seems to fit better.

Switching to simple past is, well, simplest, but the way you have this you could use passive voice in a nice way: "Thinking of ... I am reminded of".

The game has a map aroundof London on the board.

Probably.

I can see the Empire State Building as well; Iit brings back the memory of the movie called Ghostbusters (1984).

We'd probably be more likely to say "it brings back memories of".

You mean you can see it in your mind, I think? It strikes me as an unusual metaphor :) but it's not hard to understand.

In that movie, a huge marshmallow man appeared and stomped cars in athe streets.

Tense!

"in a street" feels weirdly minimalistic for the Stay-Puft Marshmallow Man. It's grammatical but it makes it sound as if he was careful to only stomp in one particular street, as if he left all the other streets untouched. I believe he was not so conscientious. More likely "in the streets".

On anthe other hand, Baker Street with Sherlock Holmes came to my mind.

"On another hand" makes it sound like you have many hands. Hindu goddess Durga would say "on another hand" :) Generally we simply have one hand and then the other. You could say it humorously if you wanted to keep an "On the one hand ... and on the other" construction going for more than two items: "and on ANOTHER hand ... and on yet another hand" etc.

I read stories of him in my childhood days and I watched movies in the 2000s too.

Unfortunately, I've never been to London, England; I've never been out of Japan, actually.

You could just say "I've never been to London". But if you want to specify the country you at least need a comma.

Someday, I wish Ito hadve a trip to London.

This is tricky. "I wish I had" doesn't really work when you cast it into the future like this - like, yes, when you say "I wish I had a million dollars" it does also mean that you'd probably be pretty happy if you got a million dollars next week. But you wouldn't say "Next week, I wish I had a million dollars". You can't exclude the present.

to-infinitive seems to fit best. But it's slightly formal, like "I wish to speak to the ambassador", as if you're taking great care to speak properly. You could say "I wish I could go to London someday", but that adds a tone of regret, it implies that you probably can't. Most natural is just: "Someday, I want to have a trip" or "I'd like to have a trip".