Are really 90 % of all foreign exchange trades dealt in USD?
The actual rate for the most recent year(2016) we have data is 88%. This comes from the Bank for International Settlements.
Why does every electronic transaction of USD legally take place on American territory? What laws/processes are responsible?
This isn't true. The US handles about 20% of all FX trades and the UK about 40%, most of which are dollar trades. However, it is probably the case that most trades in dollars could be considered regulated by the United States. This is a practical result of several sets of laws and not one specific law.
For example, the US asserts jurisdiction over the entire world for its citizens. If an American engages in some action that is legal in the country they are standing in, but not legal in the United States, then it is a crime in the US. If they commit a crime in another country it can also be enforced in the United States. For corporations, if they do any business in or with the United States then the United States acquires legal and regulatory jurisdiction. Since the banks which trade foreign currency generally have offices in the United States, they are subject to US jurisdiction.
Additionally, while there are several wire and exchange firms, they end up being required to follow US law and to interact with the Federal Reserve System if they have any nexus with America. As such anything that is fraudulent, that uses any form of wire, becomes liable under US wire fraud laws.
The statement isn't probably literally true, but since people don't have any physical control over which networks the institutions use, it is safe to assume the US has jurisdiction over the transaction along with any other countries that are directly or incidentally involved.
What about non-electronic transactions in USD?
Non-electronic transactions in paper USD do not create US jurisdiction unless one of the parties has a tie to the United States. All American fall under US jurisdiction at all times, everywhere. The only paper transactions which would have automatic US involvement would be in counterfeiting. If a Bolivian and an Argentian are standing in London and one buys a hamburger from another for a paper dollar, the US has no standing if a crime were committed.