Since orthodontics often impose many artificial little crevices and wires and generally make one more susceptible to having bits of food stuck in the mouth for extended periods of time, is it advisable to wait longer than the normal six hours between eating fleischig and milchig?
There are 2 approaches in halacha for distancing meat to dairy (see Shach YD 89:5). The Gemara says the distance is from meal to meal. Some interpret this as needing to bentch on your meat and start your milk at a different meal- the upshot being to distinguish it through separate meals.
Others interpret the gemara as waiting the amount of time sages would wait for their meal (seudas talmidei chachamim). This is six hours. While no reason is given in the gemara, rishonim posit reasons. One reasoning is that that is the time it takes for the meat particles to break down in the mouth enough to be considered unmeaty for halachic standards. After that, we negate the particles.
So, if your custom is to wait less than six hours, you must clean your teeth of any meat through kinuach v'hadacha (liquid and food rinsing) (Rema YD 89:1). If you wait the six hours, we accept the latter reasoning and don't worry about the possibility of particles (I think we accept this metzius even if we hold of the first approach to the gemara).
Even according to the latter approach, if there are definite particles in your teeth, you must remove them and then rinse your mouth even though you have waited 6 hours (Shulchan Aruch/Rema YD 89:1). You do not have to wait another 6 hours from then. You can eat immediately after rinsing (Shach there); I'm not sure how long you must wait to avoid needing to rinse.
Summary: Definite particles must be removed after 6 hours (plus a mouth rinse). Unknown particles are of no concern.
No - nothing other that the usual halachic requirements are necessary! See this recent great article on topic - relevant for Pesach too: http://ohr.edu/this_week/insights_into_halacha/5117