Melissa:
How do weak musicians learn solid timing and pitch by practicing alone at home?

As far as solid timing is concerned, a metronome can be a great help. They can be off-putting, because they're so mechanical, but to play along with one set to the required tempo can be a great help. In the heat of performance - particularly when playing instrumental music - many of us speed up, and the metronome makes us aware of this.

As far as pitch goes, well, if you have a recording of the piece that you want to sing (in the same key), stick the needle on the CD (! :D), crank it up, as Albert Lee used to say, and singalonga CD track. Another way is to record yourself singing. I hate doing this because I always detect (I think) a slight flatness in the playback, and it's true that a plain recording with no reverb and flat EQ will reveal any flaws in pitch.

If someone can't tell whether they're singing in pitch or not, an hour with a professional singing coach, or more experienced performer, can be very revealing - and helpful...