As far as I am concerned there are lots of different ways to form the negatives of such kind of sentences.

1. Bende hiç para yok. Veremem.
2. çok yemek yemiyor, ama yine de kilo veremiyor.
3. Bazı okullar burs veremeyebilir.

What is the difference in the use or the meaning of these?
I am also interested whether we can have a question form of the third one. The first two would be:

1. Gerçekten hiç para veremez misin?
2. Gerçekten hiç kilo veremiyor mu?

I suppose. But I have no idea about the third one.

Bende hiç para yok , veremem.
->I have no money ,(so) i can't give.
Here you mean , you don't have any money so you can't give money.Here "having no money" is the reason of your action "not being able to give money"

Çok (yemek) yemiyor ; ama yine de kilo veremiyor.(here no need to use yemek , its similar to English.See below..)
->He/she doesn't eat a lot(of meal); but yet can't lose weight.
Here you mean sth like : It's weird even though he/she doesn't eat a lot , he/she is not able to lose weight.
There's no reason-result relation here.(so there should be another reason for it )
(but in the first example there was a reason-result relation)

Bazı okullar burs veremeyebilir.
->Some schools may not be to able provide financial grant.

para Veremem -> i can't give money
kilo veremiyor->he/she can't lose weight
o okul burs veremeyebilir->that school may not be able to provide financial grant.

if it was sth like : o okul burs vermeyebilir.
Then our translation would be : that school may not provide finacial grant.

When you say "vermeyebilir" You mean : he/she has it ; but i don't know if she will give it or not , it's up to his/her decision.
When you say "veremeyebilir" You mean : i don't know if he/she can give it or not , i don't know if he/she is able to give.

verb + -meyebilmek/-mayabilmek is equal to may not + verb
verb + -emeyebilmek/-amayabilmek is equal to may not be able to + verb

Finally the question form you struggled to find is : Gerçekten hiç burs veremez(ler) mi?
(Actually "veremeyebilirler mi?" is the longer and maybe more accurate version ; but "veremezler mi" would be right too, especially saying "gerçekten hiç burs veremezler mi?" "veremezler mi?" sounds more natural)

veremez -> veremez mi?
veremeyebilir-> veremeyebilir mi?/veremez mi?
vermeyebilir->vermeyebilir mi?/(vermez mi acaba?)

-ebilmek/-abilmek are asistant verbs and they form compund verbs when used with other verbs.And usually -bilmek falls when combaning them into a verb.
Example : Ver-ebilir - ver-e-meye-bilir
Ver-ebilir - vere-e-mez (actually here you don't see -bilmek but it still means that way)

I hope you get the idea.

Bazı okullar burs verir mi?

Can some schools give financial aid?

Bazı okullar burs verir mi?
->Do some schools give financial aid?

Bazı okullar burs verebilir mi?
->Can some schools give financial aid?