Ommiting third person pronouns would need some more context and it has to be clear from context about who we speak because in present tense all the three pronouns on/ona/ono have the same verb form. For 1st person the form of verb is unique so adding "ja" may be somehow redundant.
Accusative and Genitive are not tenses, they are cases.
Not in all cases, no. But at the beginning of the course, almost every verb that you learn takes a direct object in Accusative.
When a sentence that used Accusative is negated, then it takes Genitive instead.
ONLY THEN. Any other case, if negated, stays the same. Negated Instrumental is Instrumental. Negated Locative is Locative. And so on. Only Accusative changes the case.