In the modern world we have electronic banking and easy transfer of assets, all done through banks, but how did deals and contracts go down in the past?

This depended on where you were in time, and as this is a 'throughout history' type question it is a bit too broad. However, most large ancient polities had a system of credit to deal with large purchases. For example, during the Roman Empire, personal bankers would control assets for the rich (landowners), who would then often lend their money out. Very little coinage ever would have been involved in large transactions, and instead, it would have been exchanges of promises. Banking systems like these (and for early China, it would have been similar) relied primarily on personal relationships. You had your personal banker store and invest money, who would in turn likely invest it through lending. A large purchase simply involved the theoretical transfer of the promise of a large sum of coin from one personal banker to another.

Would a person in Rome have had to carry large bags of coins or other assets to transfer goods, or did they use a more valuable coinage in gold?

Coinage was, first and foremost, an instrument of trade and commerce.

What happens when a ruler dies? Do the coins with that person's face lose their value and get smelt down every time for more coins? Or do they keep getting recirculated in the market along with other newer coins? What determined the value of one coinage currency vs. another, or were they of roughly the same value according to their quality and composition?

There is no hard and fast rule to answer this question, as it varied from ruler to ruler. Old coinage wouldn't necessarily be recalled unless that specific ruler was meant to be wiped out, however, most coins would remain in private hands so getting them was not a simple matter. The biggest change that occured over time in most jurisdictions was the almost inevitable devaluation of coinage through debasement. The precious metal content of coins tended to drop in time as governments in hard positions requiring capital almost always looked to debasement as the go-to strategy to quickly increase revenue.

When did currency ratios come into existence? Like here is my Song Dynasty coin and I shall trade it for a Kamakura coin 4.6 to the 1. Or here is my Macedonian coin which I shall trade for a Seleucid coin 6 to the 1.

Ancients had weights, and if the purity of a coin was trusted, that would be the easiest way to compare values. When intrinsic values come into play (particularly with debased coinage) things of course become more complicated.

And what about coins from realms without monarchs like Republican Rome? Who's faces went on the coins and how did they determine which coins got taken out or introduced to the market?

Most often gods and heroes were depicted on the side that came to be associated with the Emperor, at least in the case of Rome. The depictions of Emperors followed closely the depiction of heroes in this case (both on coinage and in other instances) which makes a lot of sense.

When did people start to understand what inflation was and how awful it was for coinage based economies? I mean you would have to lull so many coins you would literally have to hire people just to carry the bags I assume.

There was certainly some understanding of inflation, as well as at least the underlying cause of debasement of coinage. Most jurisdictions that saw coinage debasement in the ancient world also saw attempts at correcting it, often by issuing of new types of coins that deliberately set themselves apart from older coins. However, these were painful affairs.

It's important to remember that the average workers would have been incredibly poor during the ancient and medieval period by today's standards, so most people would never have a lot of coins just laying around. Hordes certainly existed among the rich as stores of value, but as mentioned above, almost any purchase necessitating a large transfer of coinage would have been done with a piece of paper.