In your first two examples, the included "that"s are being used in a relative clause. In a relative clause, the relativizer (that, which, who e.t.c) can safely be omitted when the relative clause has a non-subject gap. But with a subject gap, it's not possible. Thus, in your example, the "that" cannot be omitted because the relative clause has a subject gap(which I mark with
I'm a student that[
a student needs a car.] (The gap is in the subject position, so the relativizer cannot be omitted).
The omission of the relativer is possible with a non-subject(object or compement of a preposition) gap.
The money (that) [you are looking for
the money] (The gap or missing constituent is the complement of the preposition "for". So the relativizer is optional.)
The earrings (that) [my mom wore
the earrings] (Again, the relativizer is optional because the gap is in the position of the direct object)
In the final example:
I can't say that I'm the best challenger that has ever applied for this aid.
The first that (which you omit from the first example) can be included or omitted. This is an example of a reporting verb + that clause, where the "that" can be omitted, especially in speech or informally. In more formal English, you should probably keep the first "that".
Some further reading on omitting "that"s may be found here.