Confirmation of correct voltage and frequency to send to a speaker for it to output sound
[...]as long as the voltage supplied to the speaker measures up to
~25Vpeak from an oscilloscope, the speaker can be expected to output
sound?

Yes, you should expect sound out. With 25V peak sine wave applied, you could damage the speaker:

from overheating its voice coil wires
from distorting the voice coil form

 As Neil says, 40W into a voice coil MIGHT overheat it. It might take 40W from a sine wave generator for a fraction of a second before temperature becomes excessive.
If you happen to apply a sine wave frequency at resonance, cone excursion can be very large, so that the cone mechanically whacks against the magnet - you can hear this as bad distortion. Forces are high, and the coil winding form can be damaged. At frequencies other than resonance, cone excursion is usually smaller. Note that resonance of an unenclosed speaker (as in your photo) differs from a speaker-in-a-box.
An 8 ohm loudspeaker likely has 8 ohm impedance at ONE frequency - at other frequencies, its impedance is complex, and greater than 8 ohms.
Your voltage calculations are common (voltage is usually easiest to measure). Seems fair, because many audio power amplifiers meant to drive loudspeakers are nearest to being described as a voltage source rather than a current source. That is, amplifier output resistance is less than that of a speaker.