Consider a normal inverter which we use in our homes.
The inverter will be connected to the battery to store the energy (power), and use the same incase of main supply fails.
Now the inverter can be connected with both mains supply (home electrical source) or a solar panel via some circuit boards (charge controller).
A monitoring circuit (or switching circuit) will be there between inverter and these two power sources (mains supply and solar), it will choose which one to be connected to the inverter via electronic switches like SCR, IGBT etc.
When we have sufficient solar power then the inverter charges the battery with solar power. If there is less or non availability of solar power then it can charge the battery via mains power supply.
Incase of failure of main power supply, then inverter automatically switches to battery mode and power the house.
Some inverters uses LED indication or alarm beeps to notify about mains power supply fail status or battery low status. (that means we have NO or NC contact which we can use to show status in PLC, if required)
Well, this simple philosophy you can apply to your query, answer is simple.
Mostly we use 3 phase power supply in Industries. We have phase converters available like cyclo-converters they convert 1 phase to 3 phase, 3 phase to 1 phase or others. Each have their advantages and Disadvantages.
Coming to your queries:
Yes, UPS has batteries, so it will store the energy and use the same when main power supply fails, as like our above example. Industrial UPS are high capacity models so all design will change when you compare with house hold inverters, some additional functionality, safety, protections will add, changes from model to model.
Yes, we use batteries to store the solar power and consume the same whenever requires, aslo depends on your design.
Yes, we can interface the fault status to PLC, but its not compulsory, depends on your design again.
Regarding solar power consumption, it depends on your application and your plant standard operating procedure (SOP). Some plants use during startups, daily usage for utilities like lighting (limited), it all depends on design and procedures.
You can connect different types of Power sources to your UPS, but only one incoming power supply will be active at a time (sometimes BUS A and BUS B power supplies also considered). There may be a automatic switch or programmed-switch will be used to decide which incoming power supply will be active (depends on programming, application, usage of your equipments, or, depends on Hierarchy levels of incoming power failure), check your design or control philosophy.