Magnetic path length and Permeability relation
In a transformer, increasing the magnetic path length reduces flux and permeability and increases reluctance of the core?

True. 

The magnetic path length (MPL) is analogous to the length of a conductor for electrical current - increasing the MPL will increase its reluctance (magnetic 'resistance'). For a given magneto-motive force (MMF) in ampere-turns the flux produced will depend upon the value of reluctance in ampere-turns per Weber and so an increase in reluctance will produce a decrease in flux. (See Hopkinson's Law)

Permeability is analogous to 'conductivity' and depends upon the material as well as the path length and cross-sectional area. Increasing reluctance will decrease permeability. 

[In electrical terms if the resistance is increased the conductance is decreased]

Why does the equation of relative permeability of a gapped core show that permeability is increased as you increase the MPL?

False

The reason an 'air gap' is used is to prevent the core from saturating. {n.b. the term 'air gap' also refers to other materials such as nylon that is used to fill the gap.} In other words you are increasing the reluctance of the circuit and as we have already seen this decreases the permeability. 

As you don't show the equation you're referring to I'll just go through the basic theory.

The gap reluctance, Rg, is given by:

                        Rg = Lg / (μ0 * Ae)


where Lg is gap length, μ0 is the permeability of a vacuum (very similar to value for air) and Ae is the cross sectional area.

The core reluctance, Rc, is

                        Rc = Lc / (μ0 * μr * Ac) 


Where Lc is the magnetic core length,  μr is the relative permittivity of the core material (>>μ0) and Ac is the area of the core.

The total reluctance of the magnetic circuit, Rt, is

                Rt =  Lc / (μ0 * μr * Ac)  +  Lg / ( μ0 * Ae) 


Generally Ae = Ac = A (maintains cross sectional area across gap) so this becomes

                 Rt = (1 / ( μ0 * A)) * ( (Lc /  μr) + Lg)


If  μr is very large (as in most cases of magnetic core material) it is the length of the airgap, Lg, that dominates the Reluctance value. Reluctance INCREASES with air gap length (and MPL) and if you increase Reluctance you DECREASE the effective Permeability.