Tile – Should I replace the plywood subfloor when replacing the ceramic tile

The 45 degree decking is the primary sub-floor. It's a technique that was widely used in the past, but has subsequently been replaced with simply dropping sheets of plywood down.

In answer to your questions:

1) You can glue and screw to the sub-floor.

2) In this case, it doesn't matter. The sub-floor is carrying the load diagonally. However, if there were no sub-floor, then you would want to go perpendicular. See this discussion on the subject.

3) Always - always - always level as you go. Fix the level at the soonest possible opportunity or it can be amplified by an order of magnitude.

Two issues:

First, you need to make sure that your floor is rigid enough and planar enough for the tile to be successful. There are deflection tests to determine rigidity, and different ways of checking for planar. The bigger the tiles are, the more important these are.

1/2" plywood is unlikely to be sufficient. 3/4" plywood glued and screwed with backer board on top of it may be sufficient. An isolation membrane (I like Ditra) can help as well, as it lets the floor flex a little without popping tiles or breaking grout lines.

The second issue is floor thickness. If you go 1/4": backer board + ditra + a porcelain tile, that's about 3/4" thick. You need to set things up so the transitions to other rooms are safe; ideally they will be flush, but if not, you'll need a safe transition piece to handled the difference in level.

The installation manual for Ditra is a pretty good reference.