Read here https://blogs.transparent.com/italian/tavolo-or-tavola/ why a tavola is at the table and in tavola is on the table
A tavola meaning 'at the table .... is a similar expression to 'in cucina '..meaning in the kitchen. It refers to the table (tavola) that is the table the family eats meals from. The general table is Tavolo and this just refers to an ordinary table.. In this case you would say AL TAVOLO to express at the table.
Another native English speaker here, heard it very often! Try: https://www.merriam-webster.com/dictionary/at%20table
If you look at the top discussions in italian, I believe that is one of the topics covered. You can also read more about it here: http://blogs.transparent.com/italian/tavolo-or-tavola/
Yes. http://italian.about.com/library/fare/blfare114a.htm for a list of prepositions and what they can mean.