Luke 2:7 (the birth narrative you're referring to), likely doesn't have in view a formal inn like we would think of. The Harper Collins Study Bible (a standard in academic Bible courses at secular institutions like Yale), has this to say about Luke 2:7:

A formal inn is not in view here (as in 10.34), but rather a temporary lodging for travelers.

As implied in that note, the inn in the parable of the good Samaritan in Luke 10:34 is likely a formal inn, more like we would think of with the word "inn."

That's because the Greek terms for "inn" in these two places are different. In Luke 2:7 it's κατάλυμα, which BDAG, the standard lexicon of the New Testament, describes in this way:

lodging place. The sense inn is possible in Lk 2:7, but in 10:34 Lk uses πανδοχεῖον, the more specific term for inn. κ. is therefore best understood here as lodging.

Unfortunately, I have nothing to say about the sorts of services that would have been available at the more formal sort of inn (πανδοχεῖον) mentioned in the parable of the good Samaritan.

I can't really speak for the specific case of Joseph and Mary as I am not sure about the situation in the Levant, but in the Roman Empire as a whole there certainly were inns. These are quite well documented archaeologically in places like Pompeii, but rather than going deep into those details (long story short, an eatery with a bunch of rooms attached is probably an inn) I'll point you towards the beginning of Apuleis' Golden Ass. In this, the narrator Lucius comes upon a man named Aristomenes who talks about coming upon a friend of his named Socrates (not that one) who is disheveled and out of sorts (this sort of embedded narrative within embedded narrative is common within classical literature), and tells this story:

‘Woe is me,’ he cried, ‘I was chasing after the delights of a famous gladiatorial show, when I fell into this misfortune. For, as you know well, I’d gone to Macedonia on a business trip, and after nine months labouring there I was on my way back home a wealthier man. Just before I reached Larissa, where I was going to watch the show by the way, walking along a rough and desolate valley, I was attacked by fierce bandits, and stripped of all I had. At last I escaped, weak as I was, and reached an inn belonging to a mature yet very attractive woman named Meroe, and told her about my lengthy journey, my desire for home, and the wretched robbery. She treated me more than kindly, with a welcome and generous meal, and quickly aroused by lust, steered me to her bed. At once I was done for, the moment I slept with her; that one bout of sex infected me with a long and pestilential relationship; she’s even had the clothes those kind robbers left me, and the meagre wages I’ve earned heaving sacks while I still could, until at last evil Fortune and my good ‘wife’ reduced me to the state you saw not long ago.’

The two then go to an inn, where they share a bed (I believed this practice was common in all premodern inns). That night, Meroe (who was a witch) casts a curse on both and pees on them.

The story as a whole is worth reading for its little details about Roman innkeeping practice (as is the book as a whole, for example Keith Bradley's Apuleis and Antonine Rome is based on that). The inn serves travelers who don't want to wander the bandit infested roads at night. The owner is a woman. Entry and exit is controlled by a night porter. Crime within the inn is a concern. It seems to have been attached to a set of baths. The tale was set in Thessaly, hardly the beating heart of the empire (it is witch infested though) which argues for them being pretty common an mundane.

It is difficult to really drill down into details, but by all accounts inns in the Roman world resembled the stereotypical "tavern" beloved of fantasy.

To piggyback on this question, I've never really understood why Mary and Joseph were required to go to Bethlehem for the census in the first place. Wouldn't it make more sense for the Romans to have everyone stay where they actually lived in order to get an accurate view point of the population?