1. Depends, it's your choice. 1d10 will make an even chance of getting averages and rolls on either side of the spectrum. 2d6 makes it a lower chance for stats on either side of the spectrum and increases the chance of a more average stat, 3d4 even more.

What dice are action rolls determined by? d6? If it's d8 or higher, 5 points is probably fine.

1a. The chances of a few players getting a few stats 10 or close to 10 is almost nil, unless your group size is in the twenties and above. I suggest using

this to prevent cheating.

That, or to leave things out of chance's way, have a point-buy or template system:

Point-buy gives you a certain number of points, and allows you to 'buy' stats using the points. The higher your stat, the more points it costs to raise that stat by 1, until it caps off at a max of (presumably) 10. The template system essentially just gives you 5 different scores from high to low in accordance with point-buy and allows you to assign them to your scores however you like.

5. System meaning what dice you use for all general rolls in-game:

D&D uses the d20 system. All skill checks require you to roll a d20 and add modifiers. Attacking monsters require rolling of a d20 and adding of modifiers. Few things don't require a d20, one of which is damage, which uses the whole variety other than the d20s.

The World of Darkness and Warhammer 40k series use the d10 system. (WoD calls it the Storytelling system, but fancy names, bah.) Just about every roll requires a d10, with modifiers. Warhammer 40k requires percentage rolls sometimes, with 2d10s, one designated as the tens die, and one as the ones die.

Paranoia and the other games made by West End Games use the d6 system. d6s used for just about every roll, etc, however, there's a special mechanic with this system where if you roll a 6 on the die, it 'explodes', and allows rolling of an extra d6 to be added to the total. All 6s 'explode', which allows for whacky things like one-shotting bosses or being one-shotted youself, etc.

Just about no published system uses the d8 or d12 system, though there are homebrewed RPG systems for them. Certain published CYOA books uses 2d6s for their adventures, so there's that as well.

Also, don't use a d4 system. It's terrible and whoever who uses it is terrible.