According to Samuelson (1971) the transformation problem is to find some general formula by which transform the value of labor into market prices. This is not strictly speaking marxian problem as it already features in the classical works, especially Ricardo wrestled with this problem a lot, since labor theory of value is a classical theory but became associated with marxians later on because they derive profit from the ‘surplus value’, however empirically even industries with various different capital to labor ratios tend to have equal return on capital.
Samuelson argues there is no general solution to the problem. Since the labor theory of value fell out of favor in economics I am not sure if there are any more recent successful attempts, if there are any scholars that specialize in that here they might give additional comments on that.
The labor theory of value is not used by contemporary mainstream economists so the non-marxian solution is simply not to use the labor theory of value but rather the subjective theory of value which does not have this problem and has proven to provide better explanations than the labor theory of value.
Regarding the questions in PS:
Value is not fixed after the job is done. For example, a fashionable purse might have high value in a current year because it is in fashion but as soon as fashion changes its value might became very low. Or another example is given by 'diamond paradox'. In a normal situation diamond is more valuable than water because people already have access to high quantities of water so their marginal utility of consuming another glass of water is low, but they do not have access to many diamonds so their value is high. In a desert however the value of water would suddenly jump above that of diamonds.
No value is not generally equal to price. Assuming that the exchange is voluntary we can say that the price is either equal or greater than the value of money that buyer gives for it and the value for seller is either equal or lower. For buyers who refuse to partake in the transaction the value will be lower than price and for sellers that refuse to sell the value will be higher than price. Only in special cases is the price exactly equal to value.