Default [Q] External battery - the differences in power and charging time

Hi everyone,

Looking for an external battery to the phone I9505 Galaxy S4. I found two that I liked, but I'm trying to figure out if I can use them in any way spoil the phone.

It's about 2 batteries, which look identical (as shown below), but differ in capacity and power input:
[IMG]images64.fotosik.pl/537/97d869720d2e48d5gen.jpg[/IMG]

Code:
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Capacity: 3200mAh
Input: DC 5V / 800mAh
Output: DC 5V / 800mAh
Charging Time: 3H
Cycle Life: 500 times
Code:
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Capacity: 4500mAh
Input: DC5V / 500mA
Output: DC5V / 800mA
Charging Time: 8 hours
Cycle Life: 500 times
If any input power may cause problems?

The second thing is that on many pages I found the link that these batteries are designed for models I9500 - the point is.. if you take a percentage of these models, eg on ebay is only 1/10 was an indication of the model I9505. Is it safe to use both batteries?

The third issue is whether it is true that the longer (slower) the battery was charged, the greater lifetime retains ?

I appreciate any advice.

Best regards