In the classic example of gyroscopic precession, the wheel starts to process, and now acquires a angular moment also in the vertical direction. Initial angular momentum was in a single plane. The one due to the spin, and the one due to self-weight).
How is this new vertical angular momentum induced mechanically? Does this mean the new net angular momentum is no longer in the initial plane?
The mechanism of inducing the precessing motion is discussed by me in a 2012 stackexchange answer to the question what determines the direction of precession of a gyroscope
The start of the precession inducing conversion is that the center of mass drops. If the gyroscope wheel is spinning very fast the drop is not visible to the naked eye. Presumably this explains why the drop is very rarely mentioned. It seems likely to me that many textbook writers are unaware of the drop. Many textbook authors seem to imply that the precessing motion happens instead of dropping down. In actual fact there is a conversion (as mentioned, with the gyroscope wheel spinning very fast the conversion is near instantaneous.)
One way to confirm the drop is to try a range of spin rates. The slower the spin rate of the gyroscope wheel the larger the drop.
The drop has been verified experimentally. Paper by Svilen Kostov and Daniel Hammer: 'It Has to Go Down A Little, In Order to Go Around'- Following Feynman on the Gyroscope
Your second question was answered implicitly in the above, explicitly now:
Yes, the new net angular momentum is no longer in the initial plane.
