My expertise lies in the realm of the Byzantine Romans, so I will describe the parklands in Constantinople. Hopefully we can get a few Republican and early Imperial experts here too. If I can find any sources about the hunting question, I’ll reply further.

Constantinople’s urban landscape shifted over its many centuries of existence, but there were outdoor spaces for citizens. Several gardens are referenced in sources, including the Philopation and the Aretai (suburban parks) and the Mesokepion and the Mangana (urban parks), closer to the palace. The Philopation was just outside the city walls, and had a variety of canals, caves, animals, and other outdoorsy things. Some people living near parks grew crops in them, such as around the Church of the Holy Apostles. Over time, the city’s population waxed and waned, with some parts becoming villages within the walls — by the 15th century there were very few people left.

Also, don’t forget the Hippodrome. Nothing like a massive outdoor stadium (with occasional riots) to get the blood flowing.

For clarity’s purposes, what do you mean by freedom of movement? Do you mean legal or practical?

Sources: Henry Maguire. “Gardens and Parks in Constantinople.” Robert Ousterhout. “Building Medieval Constantinople.”

This is such a complex and multifaceted topic that instead of making a big, comprehensive post I'm going to do this in bullet point form.

• Hunting is a complex topic that varies a lot depending on who you are talking about. It was thought for a long time that the aristocrats of the Italian heartland did not indulge in this, that it was thought of as a utilitarian activity to be handled by slaves and lower status. This creates a fairly stark distinction between the Romans and the classical Greeks, particularly the Macedonians (Alexander was nearly killed by a boar). However, reevaluation of artistic and literary evidence has led to the conclusion that leisure hunting did in fact exist among the aristocrats, although it was probably not as important as in Greece. Outside of Italy it is more clear--in Asia Minor clearing the land of dangerous beats was an important status activity of the elite and in North Africa they were crazy about wild animals, although it is difficult to know who exactly was doing the hunting. There are loads of evidence in Britain for hunting, from abundant wild animal bones to popular hunting gods to signs of park management, although it is hard to know whether this was because Britons loved hunting or the relatively developed state of archaeology there.

Going down the social classes, people certainly hunted relative to their local ecology. There is a famous story about a slave in Sicily who presented the local magistrate with a boar he killed and was duly executed for possessing a weapon, as the magistrate was jumpy about slave revolt. In outlying areas hunting would certainly play an important role in supplementing diet.

• For enjoyment of nature I may be able to expand on this, but the broad answer is that many Roman villas, particularly along the coast, were built in such a way to maximize views. I can't immediately think of much in the way of urban parkland.

I'm sure you're right that the answer depended upon time, place and social class. I'm not well acquainted with the literary sources, but from an archaeological point of view, I think it's fair to speculate children would have had the sorts of opportunities to experience nature that you suggest (hunting excepted).

There are childrens' graffiti from the House of the Cryptoporticus at Pompeii, for example, that show deer, boars, and other animals - whether they are arena animals or wild beasts, I don't recall.

Martial has an epigram (XI.18) that apparently talks about window boxes: "Sed rus est mihi maius in fenestra," and Pliny the Elder laments that people used to have enjoyment of window gardens until rising crime rates forced them to shutter their windows.

We don't know a whole lot about what sorts of people were admitted into private parks and gardens in Rome - the horti, or imperial villa estates, for example. It's quite possible that common plebeians had access to parts of these estates - much like the Villa Borghese in modern Rome is open to the public.

By and large, municipal elites and wealthy freedmen in smaller cities probably had a country house in addition to their urban domus. It's nigh on impossible to link property ownership of this sort through the material record, though. Someplace like the Sarno Valley (near Pompeii) would have been quite heavily farmed and occupied well outside the town walls - that is to say, farms every few hundred feet or so.

Maybe someone else can chime in with the textual sources - I suspect there must be plenty of literary evidence...