How much authority did a Lord, traveling, like from a hunting trip (as in the beginning of The Taming of the Shrew), have over the surrounding peasants while outside his domains in Elizabethan England? Could the Lord, say, order the (female) keeper of the alehouse to do something?
I am not an expert in the period or in the history of law, and quite frankly my memory of that particular play is flawed. That said, I believe that a Lord has no legal power outside his own domain.
That said, "legal power" is probably the wrong concept - what he had was what we now term "privilege". His word as a gentleman has more credibility than her word as a peasant. (I could easily go on for a dozen pages on whether the term "middle class" is valid; I'll spare you and just use peasant with the knowledge that the truth is more complicated than the term). His desires, whims, preferences, opinions, and agenda are automatically presumed to have more validity than hers do. He has an education, greater responsibilities and an intrinsically superior aesthetic sense that permits him to understand how the world should be. Quite frankly your question is worth an upvote just for the opportunity to examine privilege and aristocracy.
Can he order her to do things? Yes - but that's the wrong question. People order me to do things all the time and I pay them all the attention that they deserve. The question is why would she comply.
She might comply because she makes her living in trade (which is another thing that makes her inferior - it is a truth universally acknowledged that if you are engaged in commerce, you have no integrity. If you sell wares, you would also sell your honor, your integrity, etc. Only a man whose wealth is secured by land can possibly be trusted). If she gets a bad reputation, then the better sort of customer will stop patronizing her establishment. (One can think of the Aristocracy as a social media phenomenon; piss off one aristocrat and you'll be lucky to be as popular as United). Also remember that the economy doesn't really work - it depends on the aristocracy to perform continuous quantitative easing. Peasants don't have ready cash (whoops, I've strayed into another area where I have to stop before this becomes a multi-page essay).
she might comply because if it is important enough, he could have his retainers take her out and beat her - or break things. She has effectively no legal recourse (the truth is much more complex, but that would require an essay on Bad King John, Tories, Whigs and the revolution of 1848. ) She can't take him to court.
She might comply because he has a position (or his family does). I don't remember the rank of the lord in question, but by definition, every lord has family with connections to government. His uncle may be the tax assessor; or a close friend may be the customs man, or perhaps the priest/deacon/vicar/. The English government at the time has a weak legislature, and the judiciary and executive branches are more important for their ability to provide loyal support to the government than for their effectiveness or integrity in providing government services. Corruption is rampant.
She might, on the other hand, have a stronger tie to her local lord, and therefore confidently tell him to pound sand. The aristocracy was not a single mind - they were political and happy to cut off their noses to spite their adversaries. The lord in question would know if the local lord were a friend or a foe, and would adjust his behavior based on how confident he was of support.
*I'm aware that this is a bad answer; no sources, lots of speculation. *
*Major hat tip to @FrancisDavey for catching my anachronism.**