Do common surge protectors prevent outgoing spikes?
Most ordinary surge protectors are designed to protect against large incoming surges only. That is surges large enough to cause permanent damage to equipment. They are not intended to prevent mere flickering of lights.
Many (maybe most) of them provide only one-time protection against large surges (e.g. lightning strikes nearby)
It is very unlikely that a typical home/office surge protector will provide the benefit you seek.
The relevant Wikipedia article says:

A surge protector attempts to limit the voltage supplied to an electric device by either blocking or shorting to ground any unwanted voltages above a safe threshold.
The standard let-through voltage for 120 V AC devices is 330 volts
MOVs have finite life expectancy and &quot;degrade&quot; when exposed to a few large transients, or many more smaller transients.

Other types of protective device can be more durable, but they still might not prevent lights from flickering.