Aspire and the park along the Corniche are well irrigated areas. I don't know where you went to parks in Dubai, but Doha is a wannabe Dubai and these two areas would compare to Safa Park etc.

It's hot everywhere in the GCC right now. From your enquiries about libraries, I'd be surprised if you found a quiet one in Dubai.

If you're looking to avoid paying for internet access at your home by visiting public wifi areas, you get what you pay for. My suggestion would be to go as early in the day as possible, no matter the venue, to avoid distractions.

Am I mistaken in getting the impression you have only ever lived in the GCC in Dubai? If so, it's the most developed area here and perhaps your expectation should be lowered - not greatly, but only by the population count and international visits.