The line between "customer" and "client" has certainly blurred, but I think there is still a leaning towards a customer being someone who purchases, and a client being someone with an ongoing, more personal relationship with the business in question, and usually more in relation to services than products. Rarely would an architect describe their clients as "customers", while a retail store describing "clients" is likely to be a more high-end store, or one that wants to elevate the value of their "customers", or otherwise be speaking in wider terms: perhaps Walmart head office would talk about "clients", while Walmart store staff would talk about "customers". The choice is a free one, but there remain subtle differences that inform the speaker's choice of wording, based on the earlier, much stronger distinctions.
I'm not sure exactly what you mean by asking "Can a customer be loyal or regular?", but I will point out that there is a difference in value again: a regular customer would be someone who frequents a store, while a loyal customer would be someone who stands up for the store, for example, they continue to frequent the store despite the store going through hard times and having to reduce stock while raising prices. This might drive away regulars, but the particularly loyal will stick by them!
A loyal customer could certainly be "upgraded" to a client, to the store owner's mind, in that there is an implied relationship between the customer and the staff, and a stronger desire for staff to "look after" the customer. A casual customer, or even a regular one, might continue to be regarded as a "customer".