Is lying flat on my back an acceptable meditation posture? Why or why not?
You can, but it's not recommended. Many people come to meditation with the notion that it is a relaxation exercise. With that preconception, it's no surprise that so many wonder why we just don't meditate lying down or in an arm chair. There may be some kind of benefit to this sort of mediation. Whatever the benefits, however, they do detract from the kind of meditation taught by the Buddha.
There is a very strong connection between the body and the mind. The condition of one really determines the state of the other. To meditate properly (from a Buddhist standpoint), it is very important for the body to be in an upright and invigorated position. When we lay down or sit in a comfortable chair, it is all too easy for our minds to go slack and lazy. These sorts of positions effect our concentration and makes it all too easy to fall into sloth and torpor. You don't want that. To hold the body upright is to hold the mind upright. What you want is a state where you are as awake (from a physiological standpoint) as you are going to get. Discomfort, even a little pain (note - little!), is very conducive to raising your attention and applying it on the meditation object.
Yes but it's easier to fall asleep that way. The four official positions are: Sitting, walking, laying and standing but really we can meditate no matter what position we are in. When we set our bodies in one of these four positions, away from noise and distractions it is like we are making a little laboratory of our mind and bodies or a little retreat so to speak but when we are off working, eating, playing, jogging or whatever, we don't have to stop and get into one of the official four positions when we remember to meditate, we only need to see things as they are.
One can meditate in any posture, but meditating lying down can be challenging because it is easy to fall asleep. Andrea Fella sometimes has to meditate lying down due to back pain while sitting; to keep from falling asleep, she holds one arm up in the air.
For the anaapanasathi bhawana "meditation that involves concentrating on inhaling and exhaling" requires sitting posture.
Only very experienced meditators can meditate lying down without visiting the sleep world. Any upright comfortable postion is good. If sleepy, imagine breathing in energy. Have a great week.