Bookstore

Every day is scorching so the bookstore becomes a good place where I am able to enjoy free air conditioning and spend my free time indulging myself in the books.

Every day (2 words) = each day
Everyday (1 word) = common, ordinary

However, not all bookstores supply consumers with chairs or sofas, so although it's cool inside, standing and reading book for a while inevitably makes my legs stiff.

In Taiwan, online stores have gradually obsolete made traditional stores obsolete and penny-ante bookstores can not compete against other famous chain bookstores.

Bookstores have started dying off here.

Perhaps it is strange but I can understand it because no one likes to keep standing.

Every day is scorching, so the bookstore has become a good place where I am able to enjoy free cool and spend my free time indulging myself in books.

The word "everyday" is an adjective, meaning common. (Things that are so common you see them every day are described as everyday things.)

You could also say that the bookstore has become "a refuge from the heat, where I can spend my free time..."

However, not all bookstores supply consumers with chairs or sofas so although it's cool inside, standing and reading a book for a while inevitably makes my legs stiff.

The word "sofa" is not used nearly as often as "couch" where I'm from, but I think that varies from one part of the English-speaking world to another.

In Taiwan, online stores have gradually obviated brick-and-mortar stores, and penny-ante book stores can not compete against famous bookstore chains.

ceaer's suggestion of "made ... obsolete" is the most common way to say it, but if you're looking for a single-word verb I think "obviated" is probably the closest in meaning. (Some English dictionaries claim that "obviate" cannot be used for things that already exist, but those dictionaries need to be updated -- things that already exist have been the primary use case for "obviate" for some time.)

Book stores have start dying off here.

You can say either "book stores" or "bookstores" (both are valid), but it's best to be consistent: say it the same way every time (within any given piece of writing).

And yes, the imperfect participle (i.e., the -ing form) of "die" is the same as for "dye". This is seldom a problem, because in context it's generally rather obvious which of these verbs is meant.