Is it true that the U.S has to run a chronic trade deficit with the rest of the world as a price to pay for being the world's reserve currency? I have heard of the argument that in order for USD to be reserve currecy, the rest of the world must own enough USD to store it as reserves. In order for the rest of the world to own enough USD, U.S has to buy more goods than U.S sells to the rest of the world. In other words, run a trade deficit.

Is this a sound economic argument? If no, what are some holes in this argument? If yes, wouldn't the trade tariffs that President Trump is imposing on its trade partners all over the world lead to the end of USD as world's reserve currency?

https://qz.com/1266044/why-does-the-us-run-a-trade-deficit-to-maintain-the-dollars-privileged-position/
This is an excellent question and I will do my best to give you a well thought out answer. 

Many observers believe the U.S. dollar (USD) will lose its status as the world's reserve currency. In fact, I got pushback from another forum regarding finance about this. These doomsday dollar folks cite agreements between China and Iran on settling trade in their own currencies rather than US dollars as evidence that the end of the dollar as a reserve currency is near.

You can actually find documentation from the IMF on what are some of the worlds currency that are being held in reserve by the planets nations: Currency Composition of Official Foreign Exchange Reservers (COFER)

Back in 2013 the U.S. trade deficit shrank to $34 billion and it was presented as good news. But if we don't run a trade deficit, that would lead to problems for us, in particular, on U.S. corporate profits. The financial media does not talk about this because it does not understand the connection between our trade deficit and our currency being the world's reserve currency which creates a dynamic that was coined in thee 60s, known as Triffins' Paradox, based on the observations of Belgian-American economist, Robert Triffin.

It was Robert Triffin who pointed out that the country whose currency foreign nations wish to hold (the global reserve currency) must be willing to supply the world with an extra supply of its currency to fulfill world demand for this 'reserve' currency (foreign exchange reserves) and thus cause a trade deficit.

So there always has to be a fundamental imbalance in the balance of payments. Depending on the goal at any given period, it may require an overall flow of dollars out of the United States, while at other periods of time, it may require an overall flow of dollars in to the United States, but net currency inflows and outflows cannot both happen at once.

How sound is this? You will have to look at it for yourself as we have been doing this for almost a quarter of a century now. So we have all been living it since the day we were born. A better question would be, how sound is it for the U.S. to maintain its dollar as the world reserve currency and run a trade surplus. That's a new one, that's one we haven't tried.

This leads us to your last question.

Just like we invoked Triffin's Paradox to explain our current situation, for this new one we can invoke Hobson's Choice.  

If you run trade surpluses, you cannot supply the global economy with the currency flows it needs for trade, reserves, payment of debt denominated in the reserve currency and credit expansion, so yes it would lead to the end of the USD as a global reserve currency.

If we no longer possess a global reserve currency, we can't print money and have it accepted as payment. (The euro and yen are quasi-reserve currencies based on the size of the European Union and Japanese economies, but neither acts as the primary reserve and as a result both are vulnerable to currency crises, despite the conventional wisdom that both are on the same footing as the U.S. dollar.)

So trade surpluses on our part are impossible as we hold the global reserve currency.