When I was a fifth grader in school, I came across a band from the UK, called the "Bay City Rollers". When I was a fifth grader, I came across a band from the UK, called the "Bay City Rollers".

Including 'in school' isn't wrong but it does sound more natural to omit it since people will assume that part.

Probably, oOne possibility is One possibility is

Probably, oOne possibility is One possibility is

The "probably" here seems unnecessary and unnatural to me.

I was really into their songs like "Bye Bye Baby" orand "Saturday Night". I was really into their songs like "Bye Bye Baby" and "Saturday Night".

I clearly remember the moment that I hadwhen I stood in front of an electronics store and had gazed at thea screen with my heart beating fast. I clearly remember the moment when I stood in front of an electronics store and gazed at a screen with my heart beating fast.

I clearly remember the moment that I had(when) I stood in front of an electronics store and had gazed at the screen with my heart beating fast. I clearly remember the moment (when) I stood in front of an electronics store and gazed at the screen with my heart beating fast.

The past perfect tense isn't necessary here. It is used when talking about a past event that occurred before another past event. However, you're only talking about one past event here.

There was the, I saw a live show recorded by the shop owner. There, I saw a live show recorded by the shop owner.

There was thea live show being recorded by the shop owner. There was a live show being recorded by the shop owner.

You don't seem to be referring to a specific live show, so the non-specific "a" would sound more natural.

Back then, video deckcassette recorders weren't common aroundin typical Japanese households. Back then, video cassette recorders weren't common in typical Japanese households.

People usually use 'VCR' since it's shorter.

Back then, video decks weren't common aroundmong typical Japanese households. Back then, video decks weren't common among typical Japanese households.

Through the glass of the storefront, I saw them on the screen wearing those lovely rRea tartan check costumes. Through the glass of the storefront, I saw them on the screen wearing those lovely Rea tartan check costumes.

I think "Rea" refers to the name of a clan...? If that's the case, then it should probably be capitalised. I'm not too familiar with textile terminology though.