In many countries you can get accounts that are denominated in other currencies. For example, Citibank does an account in the UK that's denominated in US Dollars -
http://www.citibank.co.uk/personal/banking/international/eur... - this would resolve the rate conversion issue.
However I lose a lot of money transferring money internationally due to taxes and rate conversions, and I'm wondering if I could keep the money in the US without being a US citizen, without having to physically go to a US bank.
I don't know where you are, but in many jurisdictions, doing what you're suggesting to evade local taxation could be illegal (that is, locally in terms of tax evasion. Opening a bank account overseas is not an illegal act in and of itself).
For example, let's say you're a British citizen (just because I'm familiar with the British rules, but they're likely to be "similar" across much of Europe).. if you are domiciled in the UK and earn money in the US from a Web site you maintain from the UK, you owe the UK tax authorities tax on the full income from that site BEFORE any deductions.
So let's say you make $2000 a month and spend $500 on hosting. If you set up an account in the US and then eventually only remit $1500 to the UK, you would be breaking the law if you did not declare the full $2000 to the tax man. In which case, having the remote account has no benefit except for the rates.. and you can get that locally anyway, as I demonstrated above.