I think the first thing we need to be clear about is these issues are not entirely simple or straightforward, and they can vary a little across time and place.

One could make the argument that engaging full-time as a merchant would certainly count as losing that distinction from "those who work" (a phrase used in "Strong of Body, Brave and Noble: Chivalry and Society in Medieval France" by Constance Brittain Bouchard , 1998). But nobles did engage more distantly as investors, patrons and intermediaries in commercial ventures. Chartered companies like the Virginia Company were heavily led by aristocratic courtiers at the board level like Henry Howard Early of Northampton and Henry Wriothesley Earl of Southampton, and Walter Raleigh's ventures included the patronage of Charles Howard Earl of Nottingham at times. Sylvia L. Thrupp's :The Merchant Class of Medieval London, 1300-1500" (1989) describes how manual labour was seen as the particular degrader among gentleman (gentry or aristocracy, anyone who's family own a significance amount of land and has inherited honours of some sort). Occupation of any sort was not universally taboo; governmental involvement (including earning a good part of your income from it) was never cause for degrading rank. As long as you had an inherited income and were doing suitable kinds of work.

I think it's quite possible being a full-time merchants would be considered beneath them, especially if you are driven to it out of being otherwise impoverished. But I'm not aware of any cases of that occurring or any that was specifically stated to be the case. Nobles distrusted merchants due to the suspicion of the practise of modernising commercial capitalism as a avaricious seeming thing and they disliked that merchants were competitors for social and political capital and some could be wealthier than some aristocrats. So they would feel an aversion and sense of peer pressure not to be seen to be a merchant. Which is where they might take the more normally aristocratic role as investor. And they could also seek court income, which was increasingly common in the 16th century as inflation from Southern American gold and silver made landed income unreliable as it was the most fixed source. Ironically, Constance Brouchard mentions how land's financial stability was seen as a credential, but of course that became its fundamental weakness in the Early Modern shifting economy.

I don't know whether it was the case at some point that nobles weren't taxed, but I know it definitely didn't stay that way. Regular taxation spread in the Late Medieval and Renaissance periods. Even in places like England, which did not introduce proper regular taxation until the Land Tax in the early 1700s, there were developments like the subsidy which was specifically designed to grade according to wealth and did tax the wealthy. Indeed, people below a certain level of wealth at the bottom might not pay any taxes because they had very little spare income to take. There was also the Poor Law rate earlier on as a tax that was raised on those with disposable income.

I think sometimes discussion of "taxes" in Medieval Europe becomes confused between national taxes and rents and produce dues. The dues you owed in money, and in older times goods, to your landlord were not taxes. The money is rent. Taxation can be local but is never private, even under the blurry lines of historical Europe.