Mazda 6 won't start; battery is ok & headlight is ok

Check body control module and wiring at the starter motor.

Sounds as though the battery is no good,
unless you know otherwise & also had it tested
Charging a battery doesn't address it's internal
condition
You have battery cables that corrode inside & can be
voltage drop tested
You have the starter & solenoid,ignition sw & such, to name
a few,lots of possible causes & things to check

ITS A BUTTON USUALLY UNDER THE DASH

Anything is possible, but I doubt the new battery has failed.
Chances are the alternator is not charging the battery.
The alternator should be able to power the headlights and engine once it starts.
You also need to know why the battery went dead overnight.

Can anyone guess at & list ten items?

You have to know basic auto mechanics
& have some test equip, such as a fuel pressure
gauge,volt meter & code reader.

Without factory quality diagnostic manuals to study
& follow, you will be asking for help for years.

Do you have fuel pressure at the engine?

Do you plan on buying wiring diagrams as you
progress to finding components to test in the
electrical system

You can't throw parts at it in the hope you will
get lucky,you have to develop your own test
strategy you use on all vehicles,& that requires
a great deal of theory & basic knowledge.

Faulty tai llight relay.

How about corroded battery cables (internal)

Fusible Link wiring corroded & open circuit,does
everything electrical work except the starting circuit?

Turn key & have power to starter solenoid ?
Neutral Switch !
Ignition Switch !

Tough to guess without wiring diagrams,
to see where to start testing

Can't fix what you don't know you have

Fuel pump won't be the cause. Most likely the relay.

if your vehicle has H4 headlight bulbs fitted you will find that they have both high & low beam filaments in them,what has most likely happened is that the low beam filaments are blown & the bulbs need to be replaced, also you may find that high & low beam circuits are on separate fuses, & also the front markers may have there own fuse check the fuses as well.......hope this helps......cheers.

right headlight wont work bulb is ok what to do