I've never thought he was ill or anything but an infection sounds interesting. Are there any other symptoms? or specific type of infection?
Again, I don't consider myself an expert. But, I have been keeping a number of BTAs for a couple of years. I have kept others in the past, but had never focused on them as I have been recently.
https://www.reef2reef.com/threads/a-newbie-and-his-anemones.391388/
Although I keep a humorous tone through the thread, I do take the anemones' welfare pretty seriously.
I have observed that any time there is a fundamental change in the anemones' environment (e.g., change of light bulbs, lighting schedule, nutrient levels, etcetera), there are adjustments the anemones make. In some instances, the anemones move about. In others, the anemones "purge" a lot. This corresponds to your second and third observations (i.e., closing up frequently, gaping mouth, expulsion of materials). My theory is that the zooxanthellae/internal biomaterials are being upregulated/downregulated as a result of the environmental change. As internal "stuff" is renewed in some way, the older "stuff" must be released.
If you add your first and fourth observations, the situation seems a bit more serious. These animals do like to eat. If they refuse to eat, there is something wrong. Similarly, when these animals are happy, they don't usually move around a lot. If yours is "sometimes" on the rock, and "sometimes" on the sand, there is an indication that it is unhappy. Further, these animals like to bury their feet in crevices. If they are sitting on the sand, there is something wrong. This is a bit more serious than the reaction I described above, which corresponds to your second and third observations. However, anemones displaying these two observations will also display your second and third observations. Thus, they are unhappy, and are needing to expel something.
Now, take vast experience from folks who observe gigantea/magnifica anemones that demonstrate the exact same behaviors. They are treated with antibiotics...and recover.
You can do the math from there.