An excellent (and free) Wx Radar primer published by GAPAN:
http://www.gapan.org/Papers/Weather%20Radar.pdf
Tilt management begins on page 40.
My reading material says that stabilization adjusts for the nose-up attitude and therefore no nosedown tilt is required for compensation. I assume the same is true for nose down attitudes.
Yes.
You don't mention which aircraft and which WxRDR unit you have installed, but generally -- yes.
The antenna is stabilised by the IRU (or other means in earlier units) to be stabilised with reference to the zero plane in pitch and roll.
We set 7° nose up tilt for takeoff. Is this standard for all airliners?
It is common. The 7° nose up is a reference to the centre of the beam w.r.t the plane of the earth. It is a quick way of ensuring that any beam produced by a 12" or greater dish will not give ground returns.
In the case of an 8° beam width from a 12" dish, the bottom of the beam will be 7° - (8°/2) or 3° above the horizontal plane. That is, it will not return anything lower than 320' elevation at 1nm, 3200' at 10nm, and the bottom of the beam is scanning FL320 at 100nm range.
30" dishes fitted to airliners have a typical beam width of 3.2° to 3.5°.
Setting the TILT to 7° UP prior to departure has the bottom of the beam at 7° - (3.5°/2) = 5.3° up.
5.3 x 109 feet/nm = Bottom of the beam is elevation 577' at 1nm
5770' at 10nm range
FL577 at 100nm range!
Do you adjust tilt downwards during the climbout?
Yes. I want to scan for weather, not the moon!