Is "rice", for example, considered an irregular plural?
For the more common usage (as an uncountable, or mass noun), "rice" isn't a plural, so the issue of whether it's regular or not simply doesn't arise. But when we speak of, say, different rices of the world, it's just a regular plural.
On the other hand, it's not impossible for a word which is a mass noun to have an irregular plural. In some contexts fish is a mass noun (Japan consumes much of the world's fish). And in some other contexts, it's a countable noun that's irregular in that the plural is the same as the singular (I caught three fish). But it can also be a regular plural (The parable of the loaves and the fishes).
No, it is not the case that rice has a singular form rice and an identical plural form. One way to tell is by considering subject-verb agreement. If there were a plural rice, then we would expect to see examples where the verb is plural, such as *rice are white. Not only do we not see such examples, but we find them to be ungrammatical. This same test can be applied to other nouns to discover if they simply have a plural form that shares its shape with the singular form (e.g., sheep are stupid) or not (e.g., *water are wet).
(the * indicates an ungrammatical string)