Building a beautiful tone on piccolo is a separate quest from the flute. Try placing the piccolo slightly higher on the lower lip for starters: it helps to mitigate the smaller size of the embouchure hole. Spend additional time building tonal security in the low and middle octaves before scaling the heights of the third register! I love to slur quick octaves between the low/middle registers to keep flexibility and focus...and I find singing pedal tones while playing arpeggios is so helpful to keep the larynx low. Nicola Mazzanti's book has some great written exercises for singing and playing! Whistle tones are terrific for relaxing the embouchure after spending time in the third octave. Make sure to keep the air pressure stronger than what you are used to on flute, and keep the lips a bit more relaxed than you might expect...this will give you a free and sweet tone. 'Tis the season for Nutcracker performances, with many flutists facing more piccolo playing than usual..hope this post is helpful!
Ah, vibrato...such a complex topic, since it cannot be seen but only heard. In general, if the ear is drawn to vibrato initially, it is probably too wide or too slow...a beautiful tone is shaded with vibrato, not dominated by it. In general, there should be 4-6 pulses of vibrato per quarter note pulse. Experiment with different speeds, especially on diminuendo exercises. Take the vibrato out as you go softer: next time, keep the vibrato spinning smaller and faster as you get softer. PLAY! Make your long tone exercises a laboratory for success by trying all kinds of speeds/widths of vibrato in each register and in different dynamics. Remember to keep the vibrato on the low side of the pitch so that you are not going sharper as a general trend when using vibrato.