Hmmm...
Ok, I'm actually writing up a document on this question for my own use, but its not complete enough yet to share.
I think the answer to alot of your questions depends on whether the apprentice in question is an NPC, a PC, or the retainer to a PC.
The answer also depends on whether you accept that the Leadership feat is a good idea (I don't).
If the PC is an apprentice, then he's probably bought the Patron advantage with his free advantage and we've created some patron together. The Patron is probably an average to slightly above average campaign sage in the classical mold, with a level between 6th and 9th - which would be one of the highest level wizards likely for many miles around. (Double that level for most people's campaigns. In my campaign by 9th level you are extremely powerful and feared. In many campaigns, it just means you own a bar or command some obscure outpost of 20 soldiers.) The PC will advance as fast as PC's normally advance, due to the PC's latent but prodigious talent and potential.
If the apprentice is an NPC, then he'll be somewhere in the 16 years of training which is traditional for young wizards, and likely somewhere between the age of 8 and 24 or a bit younger if he started his apprenticeship early. The apprentice is likely classless (commoner) or a 0th level wizard (see rules in the 3.0 DMG) with an XP debt of 1d100x5 before reaching 1st level, although a higher level is possible for an exceptionally talented student near the end of his training. Technically, once you can cast 1st level spells, you are free to go, though most will stay on for the full 16 years to learn as much as they can and cause no scandal.
The duties of the apprentice will vary with the master, the particular system of training the master was taught, and the age of the apprentice. The primary duty of every apprentice is to learn, and one that doesn't will likely be turned out. However, most wizards still require something of a livelihood, and so the duties may include domestic servitude, working a small farm, or helping the Master practice his trade in order to earn coin for the household. The Master's trade can very from practicing wizard, to alchemist or apothecary, to scribe, sage, or tutor. The master may also use an apprentice as an ad hoc research assistant - the classic 'stir this cauldron until I get' back or 'help me chop these mandrake roots'.
The skills a wizard requires are diverse, but mostly they revolve around teaching the student self-discipline and self-control, to cope with essential tedium, and the memorization of numerous facts. The most important skill in casting magic is total self-control - control of body, control of mind, control of voice, and control of habits. So the apprentice will be taught things to help them achieve control, from practicing being motionless in uncomfortable poses (or when struck or tortured) to dancing and singing depending on the wizard. Alot of what the apprentice does is essentially caligraphy - copying a spell over and over and over again long before he even has a clue what it means. This builds essential hand strength and penmenship. The second most important thing is the memorization of the runes and glyphs in which magic is written, and eventually what they mean and latter on still what they do. If the wizard is really good at what he does, he may even be able to teach the apprentice why they do it and the theory behind it which is something not all wizards actually know.
Wizards will also be taught things that wizards feel makes for a polished and well rounded wizard - like history, knowledge of the arcane world, knowledge of engineering, mathimatics and other practical arts other than magic, philosophy, literature, and so forth. The apprentice may also be taught various practical skills that contribute to the learning or practice of magic, like glass blowing or herbalism.
A young apprentice, certainly one less than age 9 or 10, will not be trusted to be taught magic at all - as he'll likely kill himself and others with it. Magic is extremely dangerous and difficult to control in my campaign world. It's all a mage can do not to unleash a sentient curse, conjure up something not meant to be, create a living spell, or suffer such severe spellburn that the spell sucks the soul out of their body and consumes it and they turn into a spellwight. One the apprentice has mastered sufficient basic skills, the real training begins in earnest which the learning and mastering new spells. This is an extremely scary time for everyone involved, especially when the apprentice first graduates from cantrips - which are designed to be relatively safe to cast - to full blown spells. This problem is compounded by the general precosiousness, curiousness, mischeviousness, and independent streak of extremely bright children of about this age. Hillarity of a sort often ensues.
If the NPC is a PC's retainer, then they advance at a rate typical of retainers. The easiest thing to do in RAW is simply use the Leadership feat. Personally, I hate the leadership feat as overpowered and annoying.
An apprentice is unlikely to want to apprentice himself to an adventuring PC in my opinion. There is just too much danger and too little stability in such a life. That isn't to say that it couldn't happen, but generally it won't happen until the PC settles down somewhere with a base of operations.
