LIBOR rates: Are these monthly or yearly interest rates? If I have, for instance, a Libor security with maturity of one month, with interest rate $r$. Is $r$ the amount to be paid after one month, or is it the annualized interest rate?

(My gut tells me that it is the annual interest rate, but I have not found a source to confirm this.)

And if it is the annualized rate, how is the (actual) interest payment that is due after one month calculated? Is it $1+r/12$ or is it of the form $e^{r/12}$
To answer the first part, it's an "annualised" interest rate convention - like all other quoted interest rates. For example, if a one-month money market rates are unchanged at 4%, you would receive approximately 4% in interest after a year, or roughly 1/3% a month. (Note that those numbers are ignoring compounding, further details below.)

As for the calculations, things are complicated. The high level answer is that a monthly interest rate is roughly $r/12$, with $r$ being the quoted rate. This is a simple interest rate.

The true calculations are complicated by the nature of LIBOR. Technically, LIBOR (London Interbank Offer Rate) is a polled rate of "large" banks in a number of currencies. There are similar "fixes", such as Euribor (Euro area), TIBOR (Japan), CDOR (Canada). The polled rate is based on the money market convention in each currency for a certain class of interbank borrowing.

Interest rate calculations depend upon the daycount convention, and the calendar used. If we borrow on June 1st, one month later is normally July 1st. What happens if July 1st is a weekend or holiday? There are standard comventions in each currency that determine which days are holidays, and how to move the due date relative to such dates (before/after).

There are a wide variety of day count conventions that are used across markets. I believe that act/365 (GBP) or act/360 (other currencies) are standard. (Based on this ICE LIBOR document: link.) Anyone who needs to know the exact calculations should refer their queries to their banking counterparties, as there can be a large number of small modifications to the generic description I give below.

In act/360, the amount of interest is given by the quoted rate times the fraction (actual number of days in the lending period)/360. For act/365, the interest is given by the quoted rate times (the number of days in the period/365).

Therefore, if the one-month maturity period is 30 days in the future, and the currency uses an act/360 convention, the fraction used for the one-month rate is 30/360 = 1/12. However, it is clear that the true annual compounding rate will vary from compounding $(1+r/12)^{12}$ since months are not exactly 30 days, and the number of days in the year is either 365 or 366.