Dani, I'm not sure I understand what you are trying to say in a few of these statements, but I'll do my best to correct them.
1: The first sentence is correct. The second sentence is a bit confusing. What I think you're trying to say is this: It's near the university, so I don't have to walk far or take the bus,
2. Without punctuation, this one is hard to interpret. I think this is what you mean: I didn't move to an apartment, because it's more expensive than a house, and I couldn't afford the rent.
3. This is correct, except take out "a" before fruit. You would with say I eat fruit or I eat a piece of fruit. I eat a fruit is incorrect.
4. You have an incorrect combination of a plural and a singular. Either say, "they offer fruits, but I rarely eat them," or "they offer fruit, but I rarely eat it." A native speaker would be more likely to use the singular form, but both are correct.
5. Because is the right word. The only problem with this sentence is that you need to add a comma before "and."
I hope that was helpful!