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Your summary of Marx's economics is not bad at all. For a more definitive summary I'd read Marx, Value, Price and Profit (see pdf). It's not long.
Whether or not paying someone less than the product of their labour will be sold for is necessarily "exploitation" or not is a question that is certainly up for debate, but many Marxists actually consider this to be the definition of exploitation.
However, alternatively, I will try to deal with your criticism, as best I can, by wording the economic issue of exploitation in my own way. Profit is derived from the price of the good sold, minus the cost of production (including capital, and labour - that is, wages). Or, arranged differently, Price = C.o.P + Profits. Already we see that in this instance, Price necessarily is greater than wages, since wages are included in C.o.P. If we are to expand this model across the whole of an economy, a given market, we see that on average price levels are higher than wages levels, necessarily. This means that on average, across a given economy, most people are living below the price level. The existence of private profit is the primary factor in this… and not just the primary factor. I would argue the only factor.
Now, here things get a little complicated maybe. See, C.o.P doesn't just include wages, it is wages, when you examine it to it's very root. Well - wages and profits.
Consider a product, product X. The price of X, as already stated, is determined by the cost of producing X, C.o.P, and the profits taken from the sale of X at the final stage.
But X is made up of Y and Z resources, and made with labour L and machine M - which is in turn made up of resources A and B, and so on. And the resources, they cost money because they have been extracted from the ground, or harvested, or obtained in some way. This is either more machines, or more human labour - more wages. And if it is machines, then machines made them, and so on and so on, and at its' root its always coming back to human labour. The actual cost of producing X (including all the capital, all the machines required to make it, and to make them, etc) does boil down to the wages of those who made them, the wages of those who made the machines who made them, the wages of those who extracted the raw materials, the wages of those who made the machines in order to extract the raw materials… in this economy, Price = Wages. But with profit included in every step of this process, Price > Wages.
Does that make any sense? Maybe just read the Marx, it's a different calculation but it comes to the same end.
Also, regarding your second paragraph - I am not suggesting that capitalist wage-labour is worse than feudal home-artisan trade. It is better, it is an improvement for the working class. Capitalism has been a necessary stage in human history, and has improved the quality of life for millions. And now it is time, we are ready, for the next stage. We can do even better.
With regards to scientific socialism: Of course the social sciences are not exact, they are different from the physical or mathematical sciences. They do not (or should not) pretend to be otherwise. No Marxist or socialist should ever claim to be able to absolutely determine the outcome of certain situations - it is about making educated, historical observations, and understanding the dialectical nature of humans as social creatures. Through this analysis of historical political and social movements, of class changes and economic history - dating back to the dawn of class society, at the dawn of agriculture - we can recognize patterns, see the dialectical relationship between bases and superstructures of society change, and make predictions about how human beings might behave given different possible material circumstances. Of course it is not an exact science. It is not a science of vulgar logic, it is a science of dialectical logic.