Q: Theses days と lately はどう違いますか?
A: Example Sentences:
1. Lately i've been feeling a little sick.

2. These days have gotten more harder than I thought.

These days and lately are both used differently in sentences, BUT they both mean daily or constantly, in a frequent pace.

Sorry if i didn't explain that well enough :)


Q: Theses days korean become coffee people.
A: I'm not sure I understand what you are saying, do you mean "these's days koreans are coffee people."
Q: Theses days, there's something wrong with the screen of the TV, so we have buy a new TV. この表現は自然ですか?
A: These days, there's been something wrong with the TV screen, so we'll have to buy a new TV.
Q: Theses days, I talk with my mum a lot because I'm interested in cooking and started studying cookery recently. Furthermore, my mum's food is the best. One day, I want to cook like my mum この表現は自然ですか?
A: The word "cookery" is not used very often in English. Since you mentioned cooling before, the word "it" would be more natural to refer to "cookery". For example "I'm interested in cooking and started studying it recently".
"Furthermore" is not used commonly in speech because it is quite formal so you could just say "My mum's food is the best".
The rest is perfect! :)