Power supply compatibility - 4.5v and 5v
As long as you don't select the 6, 9, or 12V settings on replacement power supply, the magic smoke won't escape. However, 4.5V may not be high enough to power up the device, in which case it either won't turn on or won't function properly, although it shouldn't cause any damage.

The replacement power supply is rated for a lower current rating, meaning that if the device draws more than 2.5A, it will brown out. This again shouldn't cause any damage. Even the cheap power supplies today have some basic short-circuit/over-current protection.

One thing to note: When you're measuring 5.16-5.18V, you're probably doing this with "no load" (i.e. when the power supply isn't connected to the device). Since the power supply is not perfect voltage source but a Thevenin equivalent, the voltage reading will drop when you connect it to the device.

Another thing to note: Most digital devices today that run off 5V feed that rail directly into a voltage regulator which drops it to 3.3V or lower. Those devices need the input voltage to be a little higher than the target voltage, so maybe 3.6-3.9V at a minimum. Bottom line, you should be fine with the 4.5V supply.