Do US authorities monitor PayPal transactions for illegal activity?

Directly? no. Indirectly? Yes. PayPal like all other businesses in the financial services sector is highly regulated on a state level in the United States with federal reporting responsibilities on transactions meeting certain criteria.

It is up to each business to set up a compliance and monitoring program, that is geared to highlight a multitude of areas such as suspicious transaction reporting, suspicious activity reporting, cash transaction reporting, money laundering flags, list flagging, and other flags that may be as a result of DD, EDD or correlation of data. Cross-examination and independent audits may also come into play here.

Sooner or later, if the account is active and illegitimate in such a way such that its detection can be done or it triggers a threshold after which further verification (enhanced due diligence) is required on the business, then this will happen at some point in time.