2) We use to talk about them by the PS3 model or motherboard model (sometimes is even neeeded to mention both) to have more precise talks, the hdd capacity is not really a good identifyer, you can see an overview of the parts and components in this tables
https://www.psdevwiki.com/ps3/Talk:SKU_Models
Your CECHH is the first PS3 model using a NOR flash type chip (with only 16mb capacity), and the PS3 firmware is around 256mb, this means a big part of the PS3 firmware files are stored in hdd... so yes... your PS3 is going to request to install the firmware after you replace or format the hdd (so have a PS3UPDAT.PUP from rebug 4.86 lite ready to install from USB)
3) I cant tell the compatibility problem of SSD when used in PS3 because i dont have any experience with them, but ive read some problems that could happen reported by reputable people. The incompatibility problems probably comes from the controller chip soldered in the SSD, the SSD manufacturers upgrades that controllers frequently so the compatibility could vary, i dont know any strict rule we could follow. As far i remember from some reports sometimes it happens that the SSD works, allows to install firmware and apps and it looks like everything is working fine but eventually it shows an error and forces you to format it (and thats an annoyance because it could make you lose all contents of the SSD)
So is a bit like what i said, if you have an SSD at home not used just try it, maybe works... but better dont spent money in buying an SSD dedicated to the PS3 unless someone reports that specific SSD model as fully compatible with the PS3
1b) take a look at ebay, i remember to see some with more capacity (but im not sure whats the max), in this section you can see the specs of some of the officials, are rated 570mAh or so
https://www.psdevwiki.com/ps3/DualShock_3#Battery
The idea of using batteries from (old) smartphones is something that has been mentioned latelly in the forum (someone even reported success connecting 2 smartphones batteries together). I have a battery in front of me from a samsung galaxy with 2600mAh, thats huge compared with the officials (but is too big, i think it doesnt fits, inside the DS3)
Anyway... if your battery is very weared out you are going to feel a huge improvement even using the officials... i cant tell the amount of hours accuratelly but overall... the DS3 i use frequently have 6 years or more and i can play for around 10 hours without recharging (i think thats good enought, 2 good gaming sessions or so)
4b) There are a bunch
https://www.psdevwiki.com/ps3/DualShock_3#PCB_evolution
In the first models they was trying different components, and in the last models they was reducing manufacturing costs too much. So the better ones are located at the center
The feeling of the sticks are good in all them are made by a company named ALPS (good quality). The biggest difference in the sticks are the pots (potentiometers) located at the sides externally to meassure the rotations in X and Y axis. The pots with 4 pins are more sensibles (and some are completly sealed, so there is zero wear out in the pots)
What i mentioned that it could be a bit tricky to desolder the original sticks in some DS3 models is because the stick have a metal body with 4 "pillars" that are intended to enter in 4 holes of the circuit board, in some of them they did this
1) Insert stick in the holes
2) Bend the pillars to "grab" the circuit board from behind
3) Solder the pillars at the back
That sucks a lot because we need to heat the solder with a sodler iron to melt it and at the same time to use plyers to bend the pillars (to make them straigh), thats the only way to remove them
But while doing it the circuit board can be damaged a bit... we are lucky there are no other copper traces around that holes though... so maybe the circuit board will have some ugly "marks" and scratches after the removal, but no problem unless you really break it
The better way to see if your sticks are soldered this way is to open the DS3 to take a look inside (and take a look also a the model of the circuit board, printed in it in white)