Learn Swedish: Jag har inte tittat på omslaget på tre veckor. - I haven't looked at the cover for three weeks.

Because the sentence is negative (har inte tittat). When you talk about how long an action has not happened, Swedish uses på for the duration. If the sentence were positive, you would use i tre veckor instead.

There are two pås in this sentence. Does that sound awkward or repetitive in Swedish?

Not at all! They are doing two completely different jobs. The first one belongs to the verb, since titta på means 'look at'. The second one is the time preposition. Having them both in the same sentence sounds perfectly natural.

What is the root word for omslaget and what gender is it?

The root word is ett omslag (a cover). Because it is an 'ett' word, it takes the -et ending to become definite (omslaget, meaning 'the cover').

Could I put inte after the main verb, like Jag har tittat inte...?

No, Swedish follows strict word order rules here. In a standard main clause, the sentence adverb inte must come between the helper verb (har) and the main verb (tittat).