What is the bandwidth of a (real) sinusoidal tone, and pulse?
The spectrum of a continuous tone is, as you said, of the form $\delta(f-f_0) + \delta(f+f_0)$: 2 impulses at frequencies $f_0$ and $-f_0$.

As a lowpass signal, this is said to have bandwidth $f_0$ (the one-sided spectrum has components up to $f_0$).

As a bandpass signal, it has zero bandwidth (there's nothing around the carrier frequency $f_0$).

If you multiply the sine wave by a pulse, this makes it time-limited, and therefore frequency-unlimited. Infinite bandwidth in theory.

In practice, you must define some criteria for estimating your bandwidth. Examples are:


3 dB drop (of the sinc function around $f_0$)
10 dB drop
drop below the noise level