Well, I can't say for certain but I believe it is "Der Mann ist am Tisch" or Der Mann ist bei Tisch". "Der Mann setzt bei Tisch" would probably be more correct if by "being at the table" you mean he ist "sitting at the table". "Der Mann setzt sich zu Tisch" would be "The Man sits down at the table" so in the motion to sit down at the table.
it would be "Der Mann ist am Tisch". "Der Mann ist bei Tisch" says "The man is by table" even beim which is the bei + Dem (by the) wouldn't be used. Beim is used more for when you are in the general vicinity of something, not right at it
see http://german.about.com/od/grammar/a/German-Preposition-Bei.htm for more info
"bei Tisch" implies that you're sitting at the table (and eating), and "beim Tisch" implies that you're near the table. I understand that "bei Tisch" may look a bit odd if you're not familiar with the expression, but it's completely idiomatic.
Yes, it is correct for every action where no movement is involved and the dative is used.
If there were a movement, then accusative would be required. For example:
„Der Mann stellt sich an den Tisch.“ This means that the man is making a few steps toward the table in order to stand at the table.
The movement can be very short as well. The fact is that there is a movement in order to place himself there. The most normal movement to imagine is that he made a few steps. He could stand up from the floor as well, come down from the table, stand up from a chair or he leaned over the table, but now he is standing there. What he really made to get there is not important and is not expressed by the verb sich stellen. Take it as an equivalent for stehen when movement is implied.
Der Mann steht am Tisch. (is standing there, he stood there 2 seconds ago too) (no movement, dative)
Der Mann stellt sich an den Tisch. (is standing there now but he did not stand there 2 seconds ago) (with movement, accusative)
Der Mann sitzt am Tisch. (no movement, dative)
Der Mann setzt sich an den Tisch. (with movement, accusative)
Der Mann liegt im Bett. (no movement, dative)
Der Mann legt sich ins Bett. (with movement, accusative)
Sorry. I wanted to write this a little bit above, and then I saw, that you are the only one, who wrote here in the last year. Maybe it can also help you.
Bei Tisch implies that you are eating. So instead of saying, I am afk in English you can say: Ich bin bei Tisch/zu Tisch. Which is both correct. Maybe you see the connection to Nachtisch, which you normally take nach Tisch = after the meal.