1) The verb be/am/is/etc. is referred to as a "copula". "be" is the infinitive form of the copula, and "to" is generally analyzed as being a particle that's part of the full infinitive (that is, "be" is the bare infinitive, while "to be" is the full infinitive).
2) "Standard" English require the copula. Forms without the copula can be considered "slang", "nonstandard dialect", "regionalism", etc., depending on one's preferred nomenclature.
Se for example https://ygdp.yale.edu/phenomena/needs-washed