Q:

What part of speech is the "be + verb" here? What tense are these sentences in?

I shall have him be killed.
She is to be stoned for adultery.

What are the constructions be +verb called, grammatically?  I feel like the above sentences are very adjectival in nature, more so than

I shall have him killed.
She will be stoned for adultery.

Is that correct?  Also, are both couplets of sentences in the exact same future tense?

A:

The construction with have (and also with make) is usually referred to as a causal construction. Then be + verb is usually referred to simply as a passive infinitive. It isn't usually necessary to invent a special term for infinitives, passive or otherwise, in causal constructions.
As a side issue, it's worth noting that if you use make, the infinitive is introduced by to when it is passive but not when it is active. So the sentnces below marked with (*) are ungrammatical:

I made him cry.
*I made him to cry.
*He was made cry.
He was made to cry.

However, even in this case, there isn't usually a special term for the infinitive: it's just a (passive or active) infinitive.

