On the first question, no (barring a sudden discovery by someone digging around in records from somewhere like Ancient Chinese or Africa that haven't yet been explored as much as medieval Europe's or Ancient Rome).
The very first collector of national accounts, in the sense that developed into our modern concepts of GDP (and National Accounting more generally), was William Petty (1623-87), an Englishman. In 1665 he published a book Verbum Sapienti, which presented his estimates of population, income, expenditure and other such concepts in an integrated set of accounts for England and Wales. The objective was to help with mobilising resources for the second Anglo-Dutch war of 1664-7. Petty then went on to publish Political Arithmetick (1676), which compared the Netherlands with France, looking to explain why Netherlands, with a much smaller population than France, was performing so well economically. The medieval period is generally dated to have ended around 1453, so this is two hundred years after.
Figures you might see in articles and books or posts on r/askhistorians with estimates of GDP per capita for medieval and ancient economies are estimates made by modern-day economic historians: the basic idea is to start off with a base level of subsistence GDP per capita on the basis that people need to be producing enough food, shelter, clothing, etc to raise the next generation (something like $250/$300 US$ per capita) and then add on based on what evidence is available about economic production. Note that GDP is a measure of domestic product, not well-being and it excludes household production of services for own use, so historians don't have to make value judgments about say the economic value of, say Maori haka relative to English Morris dancing. GDP also measures everything that is produced in a given area regardless of where the resulting income flows to, so if you are estimating GDP in Ancient Egypt while it was ruled by Rome you don't need to worry about estimating how much of their production was consumed elsewhere in the empire. The Maddison Project, an academic effort to build a database of GDP estimates, calls these sort of estimates "guesstimates". (They also have a lot more about the calculation process in articles on their website.)
As for facilitating trade, I'm going to assume that you are thinking of government involvement here, as obviously an economy facilitates trade. This raises more definitional issues, as governments are part of an economy. The Scottish economist Adam Smith reportedly said:
Little else is requisite to carry a state to the highest degree of opulence from the lowest barbarism, but peace, easy taxes, and a tolerable administration of justice; all the rest being brought about by the natural course of things.
(Source: Dugald Stewart, 1793, quoting from notes he took at a Smith lecture in 1755).
Which is a bit like saying "Little else is requisite to carry an athlete to Olympic gold than good genes, effective coaching and hard work...", plenty of countries have struggled to achieve peace, suppress corruption and maintain fiscal stability. For example The Troubles in Northern Ireland lasted for decades in the 20th century.
Determining what is and what isn't a government is also hard: in medieval England the church took on various roles that in modern times we think of as the government's role, such as determining whether a marriage legally existed (which affect property rights by affecting inheritances). Even now there are alternatives to courts for settling legal disputes such as commercial arbitration. Conversely, other roles might be filled by volunteers, e.g. farmers might be called on to form a militia where we would call in professional police or military.
We also have biases in our data sources: we have records of governments issuing orders, setting taxes, etc, but not so much of how much this was obeyed. To use another modern day example, banning the sale of drugs like weed or crack hasn't stopped their sale. People break laws and evade taxes and they don't all get caught, and if they don't then it's even harder to find data sources than it normally is for ancient economies.
And even if there is a law that's obeyed, or a tax that is collected, it can be hard to work out if it is facilitating or harming trade, there are for example lively debates about the merits of patent laws in economic history.
There are experts here on particular Ancient societies who could likely say more if you have a particular place and time in mind.
Sources
Angus Maddison,, Quantifying and interpreting world development: macromeasurement before and after Colin Clark Australian Economic History Review, Vol. 44, No. 1, March 2004
Historical National Accounting, Maddison-Project Working Paper WP-12, Herman J. de Jong and Nuno Palma June 2018, https://www.rug.nl/ggdc/historicaldevelopment/maddison/publications/wp12.pdf