[*]From the basic list of Strength, Constitution, Dexterity, Intelligence, Wisdom, and Charisma, are there any that you would like out in the next edition? Are there any others you would like in?
Nope. Those six are a sacred cow. It's possible that modules may add (or even remove) some, but the default will be those six, and only those six.
[*]What would each of the abilities mean? What would an increase of them give to an unclassed character, for example? How would certain classes particularly benefit from certain ability scores?
I would like them to get the "mental analogues" right.
Charisma is the mental analogue of Strength. It reflects your ability to influence others. And, in extreme cases (through magic), it even reflects your ability to influence the universe itself. Both Charisma and Strength are strictly outgoing stats - that is, they are used for attacks, not defences.
Intelligence is the mental analogue of Dexterity. It reflects the speed and flexibility of your thoughts. Both Intelligence and Dexterity are 'hybrid' stats - they apply to some attacks (reflecting accuracy or finesse), and they also have some defensive applications.
And Wisdom is the mental analogue of Constitution. It reflects how resilient you are, and your ability to endure mental hardship. Both Wisdom and Constitution are purely defensive - they affect saves and maybe hit points, but they aren't used to influence others.
But, apparently all stats will be used for saves in 5e, so I guess they haven't gone that way. A mistake.
[*]How much would you want the ability scores to define your character? How much should this change over time? How much should ability scores change, if at all?
Ability scores should be a distant third behind race and class. (Fourth, if we include "theme".)
Ability scores should be much less significant than they are now - the modifiers in general should be lower (Stat/3 - 3, instead of Stat/2 - 5), almost everything that modifies a stat should be removed, and they should not go up with level.
Oh yeah, I knew I forgot something. Also your opinions on ability score generation would be appreciated. I'm personally fine with most types, even the 3d6-put-them-in-order type.
The game should provide three stat generation methods, without advocating any one in particular.
1) A standard array of 16-15-13-12-10-8
2) 4d6-drop-lowest, arrange to suit.
3) 28-point buy, using the 3e stat costs.
The Starter Set should include only one of these methods, being either #1 or #2. (The standard array is probably the best choice.)
(Incidentally, the numbers on this have been carefully worked out. On average, random rolls give approximately 30.5 points, but don't allow for optimisation. The standard array, meanwhile, is a 29-point array which isn't fully optimised but guarantees decent results. And the point buy is 28 points, but allows full optimisation. Thus, there is a trade off between control and 'raw power', at least as expressed in the points.)