How does the cost-benefit relationship of owning a horse differ today from the Middle Ages or Early Industrial Revolution?

In my answer to What was the price of a horse in 1750? I provided some actual prices for horses and oxen from early Detroit, dated 1807 and 1825. If you look at the values given you find that a team of horses is worth double a team of oxen. If you look at the inventories you will also find the price of land: 40 acres is worth about the value of a team of oxen. These are frontier values, for cleared land.

I don't recall the source offhand, but a team of horses can plow more than double the land in a day as a team of oxen, but costs about double to feed, hay and oats, and horses require more care, such as shoeing. As always, time is money, and the farmer who could afford it was willing to pay more for working horses than for oxen.

In both examples the teams of oxen and horses have been trained to work together; they are priced as teams.

A horse's teeth is one way to determine its age. So you should look at the teeth if you're buying it. If it's a gift, free, then there's no point in looking. You won't lose anything if it's old, and the giver might get offended and retract his offer.

Between the start of the middle ages, until the Industrial revolution (Which majorly changed the revenue a horse could reek), what was on average the burden of owning an horse (Cost vs Income)?

The answer to both sides, cost and income, is "it depends".

Do you have a lot of land for the horse to graze on? Do you already own other horses and thus already have the equipment and skills and stables to care for a horse? Then the cost will be relatively low.

At the other end is if you live in a city. You have to pay someone to stable the horse. You have to pay for feed. You have to spend time exercising the horse. Because of stone streets you have to pay for horse shoes and additional care for their hooves. There's a higher chance of disease in the more crowded conditions.

As to income, what is the horse doing? A war horse, or riding horse for recreation, will generate no income. A horse carrying messages and fast mail might make a high income, depending on the business. A draft horse could make a living, but it depends on what it's pulling: a horse pulling a plow over a turnip field will not make nearly the income as a horse pulling a fancy carriage taxing aristocrats around.

Secondly, what would've happened if the cost of maintaining an horse was too high, but the horse would still be necessary as a ways to make a living? (They need him for transport / labout. When eating it just isn't viable)

If what you're doing for a living isn't earning you a living, it's time to do something else.

If the horse isn't earning its keep, and you're not independently wealthy, you either do something more economically viable with the horse, or you sell the horse and do something else for money.

If you absolutely must have the horse, you find a way to pay for the horse. You can go into debt, or do other work.

The people who invented this proverb had somewhat different lifestyle from yours. And lived in different environment. They worked the land. For them a horse was not a liability but an asset. And these people were the majority of population. So even if one of them had no grass to feed a horse, or no desire to work with it, s/he would easily sell it. Even if you kill a horse for meat, you get: a) a lot of meat. b) the skin from which you can make a lot of things, c) hair and hoofs etc.

EDIT. Since the question was edited, and the proverb disappeared, I give a literal translation from the Russian: "One does not inspect the teeth of a horse which is received as a gift". Explanation: teeth inspection is one of the main things when you buy a horse. But if you receive something for free, you should be glad in any case.