"Thou beest" is completely ungrammatical and was never used.
"Thou art" is the only correct form for the second person singular present tense of the verb "be".
Since that form isn't common I wasn't familiar with it (I'm not a big fan of the Shakespeare corpus, I'm more of an Old English/Middle English fan myself). I had to do some digging, but here's the result. Beest is a second person singular present subjunctive form, hence all the uses in "if..." constructions. In Middle English times, 400 years before Bill the Bard, the second person singular present indicative had already standardized to art in all dialects of English, but the subjunctive form was be (not beest, since all subjunctive person endings had been completely lost). In Early Modern English, the subjunctive mood was continuing to break down (and is virtually extinct in Modern English). Beest was a formation based on be with the addition of the (ungrammatical in subjunctive) second person singular indicative suffix -est. The more common subjunctive form continued to be be without the suffix. The form beest, therefore, cannot be used interchangeably with art (an indicative form), even if you choose to use the irregular beest. The fact that some of the quotes seem to use it in the indicative notwithstanding, the two are not interchangeable. The subjunctive was already breaking down in Early Modern English times (it was in serious decay already in Old English times and even more so in Middle English times), so the use of irregular subjunctive forms is to be expected.
I teach History of English at a university