I strongly agree about the Lifting skill. We need it. Reallife powerlifting, weightlifting training, is a skill. To allow both bonuses from ability and proficiency, better represents the wide differences between how much reallife persons can lift. Also making Lifting a skill, more clearly explains why Human females exhibit high Strength. The Strength ability includes much more than simply lifting weights. It includes agile athleticism as well. Character concepts that need heavy lifting, choose proficiency in the Lifting skill. In gameplay, the Lifting skill is important in the exploration pillar, as the old school "bend bars / lift gates", and has combat applications as well.
With regard to representing weight in game mechanics, I hate guessing how much something weighs. I strongly prefer references to size. For example, a creature can typically carry without encumbrance, a creature or object that is one size smaller. Then let more difficult challenges be Lifting checks.
I feel 5e is correct to remove the Endurance skill. It would be a crappy skill not worth choosing among the good skills. The very few times an endurance check happens, it makes more sense to check Constitution (Athletics).
Like you cannot cast in armor if your STR is lower than the Armor's base AC. Cleric and Ranger casting in armor with <14 STR should be a class based exception.
I like the idea. All armor has a Str prereq. (Or Con?) So, a character without the prereq cant successfully train in the armor.
+5 speed if you have 16 STR or CON.
I like Athletics improving speed. I worry that it is a bit fiddly. Maybe, the feat relating to speed needs an Athletics prereq.
Yes. By giving bigger bonuses for high ability score or real penalty for low ability.
The bounded accuracy of 5e cannot survive "bigger bonuses". Already, the bounds are stretched significantly − to the point that the lowest tier cant hit the highest tier. I like the situation as it is now, but creating more math disparities would break the game, especially if low level itself had huge disparities.
Initiative could be the sum of DEX and iNT.
The game math cant handle adding bonuses together, unfortunately. The closest 5e can do is granting 'expertise' or 'advantage' if sufficiently high prereqs in two factors.
CHA could grant bonus attunement slots or lower attunement if negative.
Attunement is necessary for game balance, especially to help balance casters and noncasters. To remove attunement to allow 'layering' spell effects breaks the game. 'Concentration' is narrative flavor for necessary power-gaming constraint.
The mental abilities tend to be too skewed toward magic benefits to be useful to nonmagic characters.
What if, one of the mental abilities granted a free cantrip per ability bonus. For example, Charisma +2 grants two cantrip slots? So a Fighter with the high mental ability also has some magical talent. These cantrips can scale with character level.
Generally speaking, in the same way that martial characters need different abilities for different things (like Strength for melee, and Dex for missile) I feel caster characters should as well. For example, Charisma determines the spell DC, and Wisdom (concentration, focus) determines range, area, and enhanced effects. Or something like this.