Does the capacity of the battery depend on the amperage of the memory, and is it not harmful to charge all night?

Question

Electronics

I bought a smart one after the previous one a year later (yeah, it started a few weeks after the end of the warranty) the battery died (now it works for a minute without charging, no more).
And .. immediately lost the power supply ..
It was 5V 1.35A.
Now available from the same manufacturer (highscreen) but with a current of 500mA.
And there is on 2A (china).
Now I charge from 500mA. Will this difference ruin the battery? Or is it better to use a memory with 2A? Charging speed is not important (I know that it will charge faster with 2A) It is important that it works all day later. With his family, he almost lived until late in the evening.
And the second question is, is it not harmful to leave modern smart phones on charge at night?
PS battery Li-ion

Answer 1

Better than 2A.
Not harmful.

Answer 2

And what's the difference the battery usually costs 500-600 rubles, I bought a new one and don't worry.