Я подошёл к окну и увидел, что на улице идёт град.

Подошёл is past tense, masculine, perfective: it presents the action as completed—I came up (and arrived) at the window.
Подходил (imperfective) would usually mean something like I was walking up / I used to come up, focusing on the process or repetition, not the completed arrival. In this sentence, the completed arrival naturally sets up the next completed event (увидел).

к means toward / up to and requires the dative case.
окно → dative singular окну. So подойти к окну = to come up to the window (motion toward a destination).

Not with the same meaning/structure. увидел, что ... introduces a whole clause: I saw that ... (I realized/observed a fact).
If you say увидел град, that means you directly saw the hail itself (as an object): I saw hail. That’s possible, but it’s slightly different from I saw that it was hailing outside.

Because the narrator approached the window in the past and at that moment observed a situation that was true then: it is (was) hailing outside.
Russian often keeps the weather clause in the present when describing what someone sees at that moment, especially with увидел, что .... You can also use past (шёл град) to emphasize that the hailing was occurring at that time in the past; both can work depending on style and context.

Russian typically says на улице to mean outside / out in the street / outdoors.
в улице is generally not used for “outside”; в would mean “inside something,” and a street isn’t treated as an “inside” space in this expression.