Past Tense – He {Went / Has Gone / Had Gone} Out 5 Minutes Ago

The present perfect is not permitted with ten minutes ago.

As explained here, the present perfect is a statement about the present, not about the past. Consequently it cannot be used with temporal expressions located entirely in the past, only with temporal expressions which include the present.

Ten minutes ago locates the departure in the past, and therefore requires a past tense. Since the departure was completed at that time, not continuing, a verb in ‘perfective’ aspect is required; in English, that aspect is expressed by the simple past.

If this is from an exam or class exercise you may point out to your teacher that any of the options may be correct.

I haven't seen Kumar this week. —As you say, this speaks to the current state of things: it says that you haven't seen Kumar during the part of this week that has already passed, but it leaves open the possibility that you may see Kumar during the part of the week that remains.

I didn't see Kumar this week. —You would use this if the opportunity for seeing Kumar this week has now passed: for instance, if you ordinarily see Kumar on Tuesdays, you might say this later in the week if one of you was otherwise engaged at the time of your usual meeeting.

I am not seeing Kumar this week. —You would use this if an expected meeting with Kumar will not after all take place. For instance, if on Tuesday someone asks you to bring up an urgent matter with Kumar at your regular Thursday meeting, but you know that Kumar is out of town this week, you might say this.