Battery charging for DIY laptop
1) pretty standard. Get a 4S LiFePO balancing-protection board with over/under charge protection.  About $8 on eBay. This will prevent the 4S to exceed the limit voltage, and will cut off the current if undervoltage is detected.  This board simply goes in parallel with the CV power supply and the laptop.  The supply CV voltage must be between 14.2 and 14.4 V.  You can get a 5S or 7S board and rewire it to act as a 4S board.

2) I think the best solution is a 5S 18650 LCO Li-ion (i.e. non-LiFePO).  These have more charge for the same weight and same cost.  This will work well with your 19V laptop charger.  You would need a 5S balancer-protection.  5S is 21 V, so the charger will never reach that.  If the charger goes bad, the balancer-protector (BP) board shall protect your cells.  You will not need a CV charger.

3) Yes, do not use self-protecting cells.

4) The theory is that you need a precise CCCV charger AND a balancer-protection board.  I think with a 5S BP board, you can do away with the CCCV charger and use a standard 19 V laptop charger.