Li-ion battery charging and usage
You are asking about the different ways to charge a 2S1P (two batteries are connected in series) and a 2S2P (two batteries are connected in parallel) battery pack.  Use these terms when researching this subject.

The better battery chargers will balance the charge over all the cells in a pack.  And will tend to charge the batteries faster near their charge rate limit.  They should also stop charging when the batteries are full.   Detecting the end of charge depends upon the battery chemistry and manufacture's recommendations.  

The cheaper battery chargers will not bother with balance and will (should) charge batteries at a fraction of the charge rate limit.  If they charge the batteries slow enough they may not bother with checking for end of charge.

You should avoid using linear power regulators when using batteries.  Linear power regulators regulate by appearing as a resistance between the load the battery.  It is clear, using this paradigm, that such a regulator will convert power into wasted heat.  Instead consider a switching power supply.