TL;DR : YES
You are right regarding the fact that connecting two coils of a coupled inductor (transformer) in serie makes a center tapped inductor (autotransformer).
(the words in the parenthesis are the generic terms, while the ones you use are more specific to sound/signal processing)
Why do we want gap in the core material while designing inductor?
And...
The only reason to leave gap that makes sense to is to increase the
number of design parameters to obtain a closer resulting inductance
value at the end. I can't find any other reason to leave gap.
There is a major reason and it's clear from the formulas you quote: -
What saturates an inductor is too much current and too many turns for a given core geometry and core material. However, by adding a gap we might halve the permeability of the core and this means that we could double the amps (or double the turns) to obtain the same level of saturation we had before but, the inductance will have halved when we halved the permeability.
Fortunately, when we halve the core permeability, in order to restore the original value of inductance, we only need to increase the number of turns by \$\sqrt2\$ so, if we have halved the permeability with a gap, the potential for avoiding saturation has improved by \$\frac{2}{\sqrt2}\$ = \$\sqrt2\$.
This means that you get the same inductance but now you can have an operating current that is \$\sqrt2\$ higher for the same level of core saturation when the core was not gapped.
What I understand from these two formulas is, the length of the gap
affects both the magnetic flux density and inductance with the same
proportion
And...
By leaving the gap, we reduce magnetic flux density and inductance
with the same coefficient
No; look at your 1st formula - it tells you inductance is proportional to turns squared whilst in your 2nd formula, flux is proportional to turns (no square term) so no, they don't alter with the same proportion or coefficient.
If a gap causes permability to halve, flux density also halves for the same operating current but, to return inductance to what it was previously, turns must increase by \$\sqrt2\$ hence the bottom line is that flux density has gone down by \$\sqrt2\$ for the same operating current. This is a benefit and a big one.
Best Answer
If both coils share the same core, then the length of the core is the same whether one coil is used or two (aiding). If opposed, then flux is much, much reduced.
If the current reduces because the inductance has increased then there is an overall net reduction in flux despite the number of turns doubling - this is because inductance is proportional to turns squared hence inductance quadruples (current quarters) and turns only double.
Magneto motive force (mmf) halves when the turns double because current quarters. MMF drives flux hence there is an improvement.