Q: summery haze とはどういう意味ですか?
A: "Haze" means likes a mist or atmosphere (brume/brouillard).

"Summery" means summer-like, or of summer (été).

So a "summery haze" is a misty air that comes from summertime weather.

You actually can't do this with two of the four seasons. You would say:

summery haze
autumn/fall haze
wintery haze
spring haze

Putting a 'y' at the end of some nouns can turn it into an adjective. It's hard to know what words can do it, since it sounds unnatural with a lot of words. But, do it the unnatural way in casual conversation all the time. For example, someone could say "It was like a spring-y haze" in a casual setting, but even they would know it was an odd way to say it.
Q: summery とはどういう意味ですか?
A: *summary

A summary is taking a long text and taking the important points from it. It basically gives the person the idea of what the text is about in a more condense way


Q: I like the summery sky like this. この表現は自然ですか?
A: "I like summery skies like this"

You don't need to use "the" because you're not liking just one specific summery sky, you're liking all skies that are summery like this.
Q: This summery is too long because of including surplus points. この表現は自然ですか?
A: I would say it this way:

This summery is too long because of THE surplus points


This summery is too long because IT CONTAINS THE surplus points


This summery is too long because of YOU INCLUDED THE surplus points

I hope this helps.
Q: It's a very summery T-shirts. この表現は自然ですか?
A: T-shirt should be singular be as it's is singular