What type of fiber do I need for a WS-G5484

You may have two problems.

First, the SFP is transmitting out of spec. At most, it should put out -1.2dBm per the link supplied by lacasitos above. You may be able to RMA it if a Cisco SFP is complaining that another Cisco SFP is sending too hot of a signal.

Second, because you're now getting a Rx threshold error (it should not put out less than -7.3dBm and -13.8 is quite a loss on a short link), the fiber may not be fully inserted at either the tx or rx end, or the fiber is dirty or the transmitter or receiver is dirty. (It's usually the fiber getting scratched on insertion unless you're in a dirty environment.)

If RMAing is out of the question, you could always introduce some loss to bring it under spec by using an attenuator or introducing extra patches.

PPP is a protocol for transporting higher-layer protocols (like IP) over a simple serial interface. Since you'd already be running Ethernet, PPP doesn't work and is of no use anyway. You could use PPPoE but still without gain.

Simply run Ethernet and IP on top and you're fine. All you need are routers or switches with SFP interfaces, suitable SFP transceivers and the right cable. Of course, you need to use IP addresses on the link but private addresses are free to use. Config, stir, enjoy.

In case you're worried about the security of the fiber outside a building: PPP or PPPoE doesn't help with that at all. You'd need an encrypted tunnel link, usually called VPN.

Is PPP a useful choice for fiber connections?

Not really. PPP requires an underlying hardware interface that most likely is expensive and hard to acquire - also think of a spare or replacement in a few years.

Ethernet is available in an extremely wide variety of speeds and distances and it's very low priced. By performance and by budget there's no real alternative for lighting fiber.