I'm a math nerd, so let's math this out.
You're an evocation wizard. You're going into a dungeon with a fighter. You're both level 7.
You buff the fighter with an Empowered Bull's Strength, raising their strength to 20. You buff yourself with an Empowered Fox's Cunning, raising your intelligence to 20. You also give the fighter a Magic Weapon.
Using spell reagents, you can reduce the cooldown of 4th level spells to 3 minutes each. You can only prepare two 4th level spells, so you prepare Empowered Fireballs. You also prepare one normal fireball in your remaining 3rd level slot. You have two Shields, being unable to cast Mage Armor, but you otherwise use your remaining 6 1st level slots to prepare Magic Missile, two 2nd level slots to prepare Bear's Endurance for yourself and the Fighter, and the remaining 3 slots for Empowered Magic Missile.
Over the course of a 20 minute dungeon, you go through 100 rounds of combat. You are going to use 3 of those rounds on Fireballs and 9 of those rounds to cast Magic Missile. This leaves 88 rounds wherein you're going to be using Cantrips.
The average damage of a 7d6 fireball is 24.5. Assuming you hit five targets with it, that's an average of 122.5 damage per fireball, or 183.75 with the empowered ones, for a total Fireball damage output of 490 damage throughout the entire dungeon, assuming every enemy fails a DC 22 Reflex save (13 base, 4 Greater Spell Focus, 5 Intelligence). Assuming half succeed, let's knock out a quarter of that damage, bringing it down to 367.5. Still a lot.
Your magic missiles are firing 4 missiles each for 1d4+1 damage each, averaging out to 14 damage per cast, 21 empowered. That's 84 from your 1st levels and 63 from your second levels.
Total damage is 514 from your spell slots.
The other 88 rounds you're doing cantrips for 1d4+1 damage each, or using your Evocation ability for 5d2 damage. That's an average of 3.5 per round for cantrips, 7.5 for the elemental missile. We'll generously give you 21 elemental missiles throughout the one hundred rounds of combat, assuming you use them on the first and last round. 157.5 Elemental Missile damage, 234.5 Cantrip damage.
Total damage output in 100 rounds of perfectly using something every single round: 906.
Now, let's look at that fighter.
That Fighter has a BAB of +7/+2, +8/+3 with Weapon Focus, +9/+4 with Magic Weapon, and +14/+9 with Strength. Their longsword is going to be doing 1d8+5 Strength + 1 Weapon + 2 Weapon Specialization damage, for an average damage of 12.5 per hit. We'll give them a 70%/45% chance to hit, averaging to 1.15 hits per round, outputting an average of ~14.5 damage per round.
Times 100, that's 1450 damage over 100 rounds of combat.
Let's say instead of casting that unmodified fireball, you instead cast Extended Flame Weapon on the Fighter, giving them 1d4 fire damage for 14 minutes. We'll say that'll cover 50 rounds of actual fighting. At 2.5 average damage per round, that's going to output 125 damage. About even on the fireball.
So, in situations where you know there's 6 or more targets you can hit, wizards trying to do damage can get closer to parity with fighters. It's only going to be in later levels with more scaling that they start to eke ahead.
I also don't think the cooldown in this situation would really affect much, as your spell slots are so limited that you're not going to be using many. At higher tiers, the cooldown will become restrictive. The spells you COULD cast if you could would be better than the spells you actually cast.
There is a side-effect of the cooldown change, though. A wizard is now incapable of actually killing anyone in PVP unless they use a save-or-die spell, as they can't output enough damage fast enough to beat those HP. Any auto-attacker can just hit them until they die and there's little they can do about it. There is no amount of skill that can buy you six minutes to cast again if you don't have an instant-win spell. Any damage you did, if you cast a spell and ran away, would be healed up by out-of-combat healing kits long before you could return. That seems kind of unfair to make Wizards and Sorcerers completely helpless if someone wants to just hit them with a stick repeatedly.
I suppose if I had an ACTUAL question to pose in light of all of this it would be: Will wands and scrolls be affected by these cooldowns as well?
I think this post succinctly puts into words my biggest concern with this entire change, and you havent even factored in the number of spells that require 3 and 4/CL that are meant to get stronger at CL 12/15/18.
The CL cap at its core is an OK change to make for casters by itself
And the cooldowns are an OK change to make for casters by themselves as well, albiet even if those cooldowns can be reduce by half accross the board, they still make casters effectively useless in any PvP situation, or pigeon hole every caster to focus on using 3 feats in order to increase their damage potential to get around the cooldowns (IE, stillspell, emp, and maximize to, for example cast the same spell 4 times in a row)
Ultimately placing both a damage nerf and a cooldown nerf on casters is a bit overboard and the math here above checks off exactly why it is that casters with the changes as written will have a terrible time coming up.
I do however understand the idea that CDs can be an effective way to reduce a spellcasters immense power, vanilla, in pvp. But;
In game design terms, its ineffective and misguided to do both of these.
You should want casters to fit into 1 of 2 categories here to make them worth playing, either
1. They are a big nuker with utility to support in times of need
Or
2. They have a lot of small damage over time to sustain damage while the melees go in for the kill.
In this instance, i think from talking with the Devs in discord that they would rather mages be picking and choosing the best time to cast spells rather than constantlt flinging spells.
At risk of this seeming like a whine post, ill say this:
My suggestion would be to pick one or the other
Either long cooldowns, or damage nerfs. Not both.
My preference would be, obviously cooldowns. As it requires a strategic use of spells both in and out of combat.
Mages who make it to level 18 should be deserving of that power level in itself.
As for the HP and BaB averaging. When i first read this i assumed that it meant smaller increases to both every level with sort of a reverse expontential curve to it.
So i would suggest personally a table that looks more like this :
BaB
Full bab
11 -> 11
12 -> 11
13 -> 12
14 -> 12
15-> 13
16 -> 13
17 -> 13
18 -> 14
3/4 bab
11 -> 9
12 -> 9
13 -> 10
14 -> 11
15-> 11
16 -> 11
17 -> 12
18 -> 12
1/2 bab
11 -> 5
12 -> 5
13 -> 5
14 -> 6
15-> 6
16 -> 6
17 -> 6
18 -> 7
With base HP per level up being halved after 11.
Thatd mean a barb or warden would get 6 hp base per level up
A fighter would get 5
A bard would get 3
A wizard, 2.
I think thats a better solution that still allows a bit of growth.
Apoligies for some mistakes or misspells im on mobile