Best way to power a 110VAC solenoid
Your questions can only be definitively answered by the datasheet.  Find the manufacturer and model number, then get the datasheet from the manufacturer site.

If you really need to be sure, don't buy a solenoid that doesn't come with a datasheet.

However, all that said, almost certainly a solenoid rated for "110 VAC" in the title is intended to work with line power in the 110-120 V range, like anywhere in North America.

The issue with using too high a voltage is that it will cause more heating.  The heat power goes with the square of the voltage.  120 V will therefore cause 19% more heating than 110 V.  For something like a "110 VAC" solenoid, this has almost certainly been taken into account by the manufacturer.

Your low duty cycle will also help.  Just from the picture, it looks like the solenoid won't get to steady temperature in your 5 second on time.  For a solenoid of this size, repeated 5 seconds on 5 seconds off is close to equivalent to half power.  This would not be the case if the on and off times were 10 minutes, for example.  This is just a guess based on the apparent size from the picture.

No, solenoids don't somehow self-limit the current they draw one engaged.

Again, however, only the datasheet can give you definitive answers.