Is one permitted to daven regularly at a shul and NOT pay membership dues? please explain why it is ’Asur’ or ’Mutar’ or just not appropriate.

One should pay the Shule fees as required by the Shule like any other payment he owes and needs to pay. If one does not pay his debt to the Shule is still allowed to enter and daven there. If the Shule is a private Shule [i.e belongs to a private group and not a public place open to all like the Kotel for instance] the owner / partners / members can prevent entrance to whom they wish to. [Rema OC 153: 16] If your Shule is such a Shule you may not allow such a person to attend Shule services [see Biur Halacha ibid that the above may apply only as long as there is another Shule he can attend] Similarly you can prevent him from getting an Aliya to the Torah if he insists not to pay. [Mare Habazak 1: 11] In the case he is unable to pay for financial difficulties it will be appropriate if the Shule would try to accommodate his special situation.