Hello, so from my point of view I would say that the app is trying to install an important update. The first thing I would try is to do a hard reset. If it doesn't work, maybe contact support. But it should actually be possible to start the AP again. Provided, of course, that the power supply is OK.
This could be because the switch is supplying too little power for the AP... To be on the safe side, you could install the injector, then the AP will certainly have enough power
No, since the U6-LR needs 802.3at, which the U-POE-AF can not deliver.
You can use the POE-48-24W-G, which is listed by Ubiquiti as a compatible POE-injector for the U6-LR.
yes, it works. the EAP245 supports 802.3af: https://www.tp-link.com/uk/business-networking/ceiling-mount-ap/eap245/#specifications
However, the EAP245 should actually already come with a power adapter.