Last year, I bought a Gree brand air conditioner, and installed it in one of my houses.

This air conditioner was boughtpurchase online from the Gree flagship store.

a flagship store generally denotes the largest store of a chain. For instance, the flagship store for Louis Vuitton in on the corner of Champs Elysee and George V in Paris. I'm not sure if it is used to describe online purchases.

After a period of time using itusing it for awhile, I found some problems with it.

Firstly, no matter at what temperature youit is set, it wais always working all ton and never stops even when time, and never stopphe desired temperature is reached.

Secondly, its heating or cooling effectfeatures are not good.

You can be aware of thistell when you put your hands on the air outletin front of air vents.

Gree is the topmost popular air conditioner brand in China, and it is very popular in China too.

It should accounts for 80 percent of all the air conditioner shares.s sold.

Is the product quality of an online and physical stores differentproduct different from the ones in the store?

Or is it just that this one that has quality problems?

you can also say idiomatically, "or is this one just a lemon?"

In a word, just know that I am very dissatisfied with itby its performance.

This air conditioner wasI bought it online from the Gree flagship store.

Your sentence was grammatically correct, but had two issues:
(1) It was in the passive, which sounded awkward because you were talking about your own actions, not some unknown person.
(2) It repeated "air conditioner", which was unnecessary.

After asome period of time [using it], I found some problems with it.

The phrase "using it" is unnecessary.

First[ly], no matter what temperature you set, it was always working all the timerunning, and never stopped.

"First" sounds more natural than "firstly".

"Working" can mean either "running" or "not broken". I used "running" to make the meaning more clear.

Secondly, its heating or cooling effects are not good does not have much effect in heating or cooling.

You can be aware oftell this when you put your hands on the air outlet.

Gree is the top air conditioner brand in China, and it is very popular in China too.

It's clear from this sentence and the next one that the brand is popular, so there's no need to write that.

It should[probably] accounts for 80 percent of the market share of air conditioner shares.

"Should" can imply obligation. In this case, "probably accounts for" is better.

I am very puzzled aboutby the performance of this air conditioner.