“these days” is correct; “this days” is wrong. The demonstrative determiner “this” has a singular form “this” and a plural form “these”, and its plurality must match that of the noun that follows it. “days” is a plural noun. These days is correct.
Correspondingly, is nowadays grammatically correct?
The correct word is "nowadays." "Nowdays" and "now a days" are errors (although the correct "nowadays" comes from a Middle English phrase that was written as two or three words.)
Which is correct this past few days or these past few days?
Past simple: These past few days were fun. Present simple: These past few days seem like ancient history now. Present simple: I often replay these past few days in my mind. Past perfect: These past few days had been the worst ever, until today.
What kind of word is nowadays?
Nowadays is the only correct spelling of this word. Spelling the word as three words—now a days—is incorrect.
What did you do this past few days?
It is a present perfect tense, and as long as the time phrase matches the tense time period of 'since then to now', the result should be OK. So when we say 'the past few days' we mean the complete set of 'past few days', not just some of them, and which means this time phrase matches the tense.