how many half notes, and in which direction to transpose tenor to alto sax?

thanks for any help

The tenor sax is B Flat, and the alto sax is E Flat, so five steps up.

Can't help with your second question.

Yes, Barry is correct, and it is half steps, not half notes.

"Five steps down would be an F, right?"

Five steps down from What?

Look at it this way.
When the piano plays middle C the Tenor plays the D 14 semitones above.
When the piano plays middle C the Alto plays the A 9 semitones above.
All instruments are sounding middle C but the written Tenor note is D and the written Alto note is A (written 5 semitones below the tenor).

"relative to C"

Yes it is but it works the other way.
When the Alto plays C the Piano plays the Eb below.
When the Tenor plays C the Piano plays the Bb over an octave below.

The transpositions from written piano to:-
written Alto - up 9 semitones.
written Tenor - up 14 semitones.

From written Alto to written Tenor - up 5 semitones.
From written Tenor to written Alto - down 5 semitones.

The Alto is pitched higher so its written lower.
The Tenor is pitched lower so its written higher.

It's confusing unless you play both.