Yes. Every row and column completed by a placement is found first and then cleared at the same time, so crossing lines both go.
Yes. After you place a piece, the game finds all the completed rows and columns before it removes any.
Why that matters
Because lines are detected first, a row and a column that cross at the square you just filled both count. Clearing one cannot spoil the other. Every cleared line scores its 10-point bonus.
The square where two cleared lines cross is shared, so two lines free 15 squares rather than 16. Three lines (for example two rows and a column) free 22.
What does not clear
Only full rows and full columns clear. Diagonals, blocks and partial lines stay. Cleared squares become empty at once, so a piece from your tray may now fit where it did not before. Look at what the clear opens up, not only the points. The strategy guide covers planning crossing clears.