Wow learning curve... Anemones do not have venomous stings like bees. Instead, like corals, anemones have nematocysts in their tentacles. Nematocysts are small vescicles packed full of POWERFUL enzymes within the cells of the anemone (especially at the skin) that serve two functions.
First, they are capable of "grabbing" things the nem wants to kill/eat. People refer to the tentacles of a healthy anemone as "sticky" because the nematocysts will try and stick to your fingers if you touch them. This is the nem's way of saying, "Stop touching me or I'm gonna eat you".
Their second function is in some cases digestion. If the anemone contacts small prey like copepods, phytoplankton, scraps of dead fish/shrimp, the nematocyst will first grab hold of the small particle, in the process breaking open and releasing digestive enzymes that literally dissolve the prey directly at the nematocyst where the cells of the nem absorb it. If the prey is too large, the nematocysts hold on to the prey as the tentacles draw it into the nem's mouth for digestion by its stomach.
What does this mean to us? Well, some anemones are "stickier" than others. The species I mentioned earlier are VERY sticky. So sticky in fact you can often rip of the nem's tentacles if it does stick to your hand. BTA's are some of the "least sticky" anemones, but keep in mind they should still be sticky. Its kind of like saying a car with a 4-cylinder engine does not have a lot of power. Sure it's not gonna break 1/4 mile records, but it will certainly get you going somewhere quickly if you push it. Unhealthy anemones are NOT sticky. Many people will use stickyness at the LFS as one of many gagues of the anemone's health before purchasing it. Be on the lookout for those which do not grab hold of fingers. Ask to touch it, or ask the shop owner to, it should stick to a person's fingers.
As for the mouths, yes they have a mouth in the center of their body. The size of course depends on how big the anemone is. However, an anemone's mouth should never be "gaping" open unless it's about to eat. The mouth should remain closed and pinched shut. If the mouth is gaping open, this is the first sign of poor health. If the conditions leading to the nem's stress/poor health are not removed, the anemone may literally turn itself inside out and push its foot through its mouth. If this happens, the nem is probably allready dead. I've never seen one recover from turning itself inside out and other far more experienced reefkeepers than I claim they've only seen it a handful of times. So again, beware of this.
BTA's are widely regarded in the hobby as being the hardiest of anemones, but again, they are still difficult to keep. Rather than asking a lot of these questions, I'd suggest you concentrate on the methods of water husbandry and equipment selection to keep an anemone alive and happy for your tank should be at least 6 months old before even considering one.