If "to be" is the copula, is there any special name for "to have"?
The term copula is used in more or less strictly defined ways:
In English, the verb be is sometimes referred to as "the copula," but other verbs (identified in Observations, below) have a copular function as well.
copula ... 1. (Linguistics / Grammar) a verb, such as be, seem, or taste, that is used merely to identify or link the subject with the complement of a sentence. Copulas may serve to link nouns (or pronouns), as in he became king, nouns (or pronouns) and adjectival complements, as in sugar tastes sweet, or nouns (or pronouns) and adverbial complements, as in John is in jail
[Collins English Dictionary – Complete and Unabridged © HarperCollins Publishers 1991, 1994, 1998, 2000, 2003]
cop·u·la a connecting word, in particular a form of the verb be connecting a subject and complement.
The last, loosest 'definition' certainly allows verbs like blush in 'the rose blushed pink' (which has semantic weight and which I'd call a 'link-like verb') - and possibly coordinators and prepositions. I don't recommend it. The other two usages are accepted (and partly contradictory).
Other incarnations of 'be' are as main verb (I am) and auxiliary (I am writing). 'Have' has incarnations as a main verb (I have enough money for a pint) when it is sometimes delexical (I'm going to have a bath) and an auxiliary (I have cut myself). But then so does 'do'. (Sundays, I do for Mr Jones / she did a jig on the table / I did go).
Have is one of the three English primary auxiliary verbs. The other two are do and be. When used in constructions with other verbs, they show how the main verb is to be understood.