If you're handy at all repairing this type of faucet is typically not a big deal. I'm assuming from your description that the cold water side works.
As you indicated, it could be calcium build-up but it might not as well. The problem is almost always with the cartridge and occasionally in the shutoff valve. Turn off the water supply to the faucet. Hopefully there is an accessible shutoff. With the water off you should be able to remove the handle and any retaining nut that would be holding the cartridge in place. They're all a little different so watch carefully how you take it apart to make sure you get it back together properly. Sometimes the cartridge will unscrew. Once it's out the best approach is to replace it. Usually your local home center will have one. However, check it for debris. I've had situations where the washer in the shutoff valve had crumbled and pieces of the washer were blocking the flow. If that's the case come back and someone will walk you through how to repair/replace it. I've included a picture of a Moen cartridge for a roman tub faucet so you know what it looks like. Yours may be different.
First, are you on a well or on city water? Wells sometimes kick up sediment that can get into the water system.
Obviously, debris is coming from somewhere in your water supply lines. I can't tell you specifically where but I can possibly help you track it down. Since it's occurring at several locations throughout the house the problem must be somewhere in the main water line at some point before the pipes split off to supply those faucets.
- Check all of your faucets by removing the filter screens on each faucet. Even if they're not totally blocked you may see some of that black sediment. Make note of those faucets with the sediment and those without.
- Check the black sediment. Can it be rubber pieces or is it like sand? If it's sand it probably is coming in with your water supply. If it's like hardened rubber it's probably caused by something in your water system.
- If only some faucets are affected the problem will be somewhere on the line before those lines tee off to their respective rooms. Check any of the shutoff valves on that line by turning them off to see if they stop water flow to the faucets. If water continues to seep through when it's off, and that valve is a washered valve, it may be part of the problem and needs to be replaced or repaired with a new washer.
- If all faucets have some of the black sediment the problem is somewhere along the main water line or even with the water supply itself.
- At this point check any in-line devices you have including filtering systems, water softeners, RO systems, etc. These systems have gaskets and filters that if not properly maintained can cause rubber parts to disintegrate.
- Lastly, check your main shutoff using the method described above. I doubt the problem is at the main shutoff unless it is a washered valve. It will probably be a gate valve or ball valve neither of which has rubber washers.
Hopefully, this will help identify the cause - if not, you need to call your water company and see if they've been doing any repair work in the area. Good Luck!