Miriqua was literally made for this (among other things). She'll keep up with it to the extent of their stamina.

Outside people are bustling around distributing goods and services and social interactions to one another.

They can pick up on bits and pieces of conversations! Not as many as they could have in London, especially when the goods and services and social interactions are ones they don't have all the context for, but they're making progress. 

As one pays attention one becomes aware of a pretty bright line between proles and everyone else. There are more proles than there are everyone elses, but the latter are more attention-getting, have more and livelier conversations, once a pair are glimpsed to have a child with them, they have more complicated social relationships. The proles have work, they mostly talk about work, they mostly socialize with either people they know through work or the geographically nearest prole socialization locale. It's not that they're dim or lifeless. They are bright, lively people who, as it happens, really enjoy making shoes or sweeping streets or cooking meals or doing laundry for fourteen hours a day.

They both love their work, but only what feels like a reasonable human amount; the proles are a bit unsettling. How much time do the other type of people seem to spend working?

Varies widely. Non-prole work is less suspicious to be really fascinated by, anyway - research, the arts, governance and diplomacy. Some professions seem to have a mix of prole and non-prole practitioners, including medicine. It seems perfectly respectable for someone understood to be well-made but not aimed at anything that would (in a society with money) constitute gainful employment to bop around trying various hobbies and dabbling in things while living off the delightedly-provided prole labor.

Watson would be far to embarrassed to do that sort of thing himself if there's any way he can get gainful employment instead, but Holmes will appreciate being able to continue his Bohemian lifestyle of research and the violin. Or more likely some novel foreign instrument, since finding an actual violin here seems implausible.

There are novel foreign instruments to be had, though they seem to be mostly custom made rather than kept in stock; maybe he could get a violin if he described it well enough.

He can definitely describe a violin well enough, and will probably do that if it's that easy to get something custom. He will, however, resist the temptation until he has slightly more permanent lodging.

Watson, meanwhile, will begin a search for a medical teacher. Not a particularly systematic one, just yet; more exploring the ways in which teachers and students might find each other. Do people post advertisements for that sort of thing, for instance.

Generally not. It's typically more of a word of mouth sort of thing.

Then if he hears anyone talking about medicine or visiting a doctor, he will inquire of them in halting vernacular.

This person is on painkillers for his shoulder injury.

"Excuse me, sir, can you tell me what doctor you went to?"

"Oh, yes, I saw Beretn."

"Thank you. And can you tell me where is--where he works?"

"The corner of Sixteenth and White Street."

"Thank you again!" It may be too late in the day to head straight there, but Beretn will probably be there tomorrow. For now he'll go back to language lessons. 

There aren't really days. People seem to have loose schedule syncing, but there are suns in all directions as the round turns.

Yes, "tomorrow" is more of a shorthand for "after they've eaten again and studied some more and had some sleep" than any well-defined astronomical situation.

When those conditions have been met, Watson sets out for Sixteenth and White Street, having obtained directions from the innkeeper and signage transcriptions from Miriqua. Holmes wishes him good luck on his way out.

There are of course four corners to the intersection, and a few buildings in all directions that could be described as "on the corner". That one over there has a ramp, though, and none of the others do.

Then that's the one he'll knock on first. 

"Come in!" calls a voice.

In he goes, and takes a look around.

It's an office, but there's a hallway off which what might be patient rooms branch. An elderly woman is manning the desk. "Hello, what brings you here?"