My understanding of the 6th precept is that one should only eat between sunrise and noon.
Yes. (if one is very interested, there is a very informative work about the beginning time: When Is Dawn (aruna)? When Is Dawnrise (arunuggamana)? - Bhikkhu Ñāṇadassana for astrological and people interested to be strict and informed)
My question is twofold.
Firstly, what is the purpose of it. Is it to practice disciplin, to renunciate pleasures and so one can gain more certainty in their ability to be diligent? Is it so that one knows hunger and does not act upon it?
The reasons are many. Yes, the 8 precepts are to reduce sensual pleasure which is the biggest obstacle for meditation.
While doing so, one is naturally forced to keep mindfulness because otherwise he/she would be overwhelmed by sense desire and could not keep the precepts. That means observing them is already much about sati (mindfulness). The next reason is reducing food which means one does not like to harm for just pleasure. This is the metta and karuna aspect of "fasten". A further aspect is the time aspect. One safes a lot of time which are normally used to search and organize food, prepare earn money ... with all its dangers to break basic precepts.
And yes, it is also to learn about dukkha which one does not know when just eating when hunger arises or sense desire. Yet one also learns to know the different between bodily hunger, hunger after pleasure and also what the body needs.
One more reason is that normal people have the most pleasant food in the evening, so one needs to let go of such as barbecue and dinner and such things with all its other most not so good involvements.
Secondly, if one is taking the 8 precepts, they should take all 8, right? There is no "pick and choose the most convienient". This sounds silly when expressed, though i've read statements on buddhism.SE and in books saying "don't do X if it would cause major conflicts in your life and you don't value X over the absense of conflict", be it child raising, vegetarianism while dependant on ones parents, meditating more at the expense of getting sufficient sleep, etc.
Silly goes most again clinging on something. There is no such as keeping precepts without giving up and letting go. BUT, it really no problem to start with one precept and step to it and then maybe the next. One would not be called a person who keeps 8 precepts, but that should not be important. Important is that one sticks to his/here resolves. It might be that one has a job where he/she needs to be attractive and is not able to let go of that the specific silas demand in his/her time of work and for now there is no other way. So its really better to make such compromises as to say "I can not keep all". Better 6, 7, 7.5 strict that 8 just literary.
I still eat meat (and dinner) due to family pressures. Is it acceptable to make these allowances?
Its no fault and not unwholesome to EAT meat. One does good not to demand it, not to speak about bringing people to organize it or even being involved in killing. That is a thing that comes step be step. Even in a family members start to adopt good behavior and will respect if one members does not like to be involved in demanding meat. But again, there is no problem to eat meat. Just do not develop joy when people still stick to it and maybe give them your portion if they can not let go of desire for meat and eventually problems with it.
My concrete example is that I often stay up very late when it is the days surrounding the New Moon, for Astronomical observation. (This is a date that many Thai laypeople commit to the precepts). So leading up to that day, I gradually stay up later and wake up later. Eventually, I am waking at noon.
If you keep the Uposatha, its usual for mediator to even do not sleep the night before. An its no problem to let go of one night sleep.
Uposatha can be hold (just) on the Uposatha days and Sila days (7 or 8 day after full and half moon) some give it some additional days.
What are the different categories of observance?
The times for keeping the pakati-uposatha are the 5th, 8th, 14th and 15th of the waxing moon and the 5th, 8th, 14th and 15th of the
The times for keeping the pati-jagara-uposatha are the five days of the waxing moon, i.e., the 4th, 6th, 7th, 9th, and the 13th, and
the six days of the waning moon: the 1st, 4th, 6th, 7th, 9th, and 12th
or 13th. That makes eleven days in a month for the observance of this
type of Uposatha. ... see more: Uposatha Sila
And then there are people who make it akaliko, so with no difference of day, to keep the eight precepts which is normally possible for everyone and most suggested, especially of those who strive for an end of suffering and to go the upward path. No gift is great as this for all being of this world like the gift of silas, the Uposatha-precepts.
Atma has to say that he is not that happy with this very traditional and formal explaining which has adopted much from the Vinaya and much which is out of context. Also Atma is not happy about the factors and many more. So generally its good to stick with the suttas:
Muluposatha Sutta (AN 3.70) — Discourse on the Roots of the Uposatha
Uposatha Sutta (AN 8.41) — Discourse on the Uposatha Observance
Visakhuposatha Sutta (AN 8.43) — Discourse to Visakha on the Uposatha
Sakka Sutta (AN 10.46) — To the Sakyans (on the Uposatha)
Also DN2 give much info about virtue of a eventually yogi. For family life maybe to much.
It is impractical and unhealthy at these days to be eating at sunrise, only to fall asleep, and then fast until I return to sleep the next day. Should I just fast for the same duration, shifted to healthier hours?
No. If you need that sleep. Eat breakfast, noon. If you than really need to rest, there should be no problem.
More infos Upasaka might find here: Uposatha Observance Days and also a this years calender
If more question arise and certain which for discussions appears, Upasaka and others are always invited to make use of the online monastery sangham which might be in some cases maybe a better place. This environment is certain limited and for some issues not so proper even if most helpful and supportive in many aspects. So it the "invitation" should be not misunderstand.
(Note: this answer has not been given with the agreement to be means of trade or the purpose of/for trade and/or keep people trapped and bound. How you handle it lies in your sphere, but does not excuse the deed here either.)