Monetary Approach to a depreciating currency
Let us call the country with the trade deficit, country A, which is our domestic country. If a country builds up a trade deficit, it is selling less goods to other countries than it buys from them. To do so, A must exchange its currency for foreign currency. This means more demand for the foreign currency and more supply for the domestic currency. More supply reduces the price of domestic currency relative to other currencies. More demand for foreign currency increases their price relaive to domestic currency. All this causes a depreciation of the domestic currency. For more detail on this see, for example, the answer to this question.
Currency values are fundamentally determined by supply and demand for currency (see answer above). Both of those are driven by both the goods and financial markets. So trade deficits have an effect here, as well as investors flocking to higher rates.
Foreign investors both affect and are affected by the exchange rates. If these investors want to invest in a foreign country with higher interest rates, they need the currency of that country in one way or another. So they sell their currency to buy the foreign currency, which like in the answer to 1) will have an effect on the value. Even if foreign investors have the foreign currency to invest, they will need to exchange it at some point in order to consume their earnings at home.
Solid empirical evidence is rather difficult when it comes to long term effects. It is true that the US has had a trade deficit for a long time. However, it is unclear how long a time is a long-run. Furthermore, before having a deficit, the US has had a surplus for a very long time, so it is natural it would eventually have a deficit. Whether this deficit has exceeded past built up surpluses is not necessarily clear.