Your questions:
1. Using a collet is a lot faster than dialing in on a four jaw chuck. If you are making a lot of parts with the same OD that will fit in a collet, you will save a lot of time using a collet. That's the why.
2. I do not know the general nature of your work so I will not make a suggestion. The 5C collets are more limited in the range of the size of the part that they can hold. A 1/2" collet will only hold a part that is within a few thousandths of that size. So you essentially need a collet for every size you will need to hold. On the other hand, the ER collets have about a 1 mm (0.040") size range and a set typically will have overlap from one collet to the next. So, a set of ER collets will hold any diameter from the lower end of the smallest collet in the set to the upper end of the largest one. But, ERs are generally only available with round holes while the 5Cs can be had with square, hex, and probably other shapes.
Oh, and there are a lot of collet holders for the 5Cs that allow fast, lever operated operation so you can change parts while the lathe is still running. This saves a lot more time in production situations. The ER collets are generally tightened with a special nut in front of the collet and you must stop the machine and use a wrench to loosen and tighten it. So it will be slower.
Your choice, not mine.
3. Most mills have a collet chuck that is either built in to the spindle (R8) or made specifically for it. Interchange between a mill and a lathe would be difficult. I have never seen an R8 collet used anywhere but on a mill. The only collets that I have seen used on different machines are the Morse Taper collets, but they are out of date and hard to get. Work holding in the lathe and tool holding in the mill are two different things and each has it's own problems. If you pursue this, do be careful of all the details before purchasing anything. I understand the temptation to try to economize, but I do not recommend it. It will be more trouble than the savings, if any, are worth.
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Before you start looking for collets, etc. Get the number of the machine and post it up. Then it will be a lot easier to steer you in the right direction. Although what everyone has posted is great advice, I can see where you would get pulled in 20 different directions at once. There are many collets that you can get. By that I mean not just for the machine, but for drilling and end mills. You can get straight shanked drill chucks to fit in a mill collet. You can also get a straight shanked collet holder that fits into another collet to run smaller end mills. But what you need to know is exactly what collet fits your machine, so to give you that, a number from the machine would be a huge help. Then we can have you cutting small ones out of big ones in no time.
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