1. Two-thirds of the apples were bad.
2. Two-thirds of apples were bad.
3. Most of my pens are expensive.
4. Most of pens are expensive.
5. The rest of those boys were nervous.
6. The rest of boys were nervous.
(Do we have to use 'the' or 'those' or 'my' before nouns?)
7. Half of some pears were rotten.
8. Half of some water was thrown away.
(Are both grammatical?)
1 is fine.
2 implies two-thirds of all the apples in the world, which doesn't make sense.
3 is fine.
4 would be fine if you omit "of."
5 is fine.
6 is not. Yes, you need some kind of identifier before the nouns in these phrases.
7 and 8 are grammatically correct, but quite vague and not useful. The word "some" makes me want more information.