Learn Dutch: Gevonden heb ik de fietspomp niet. - Find the bicycle pump, I did not.

Moving a past participle to the front of the sentence is used to create contrastive focus. By starting with Gevonden, the speaker emphasizes the action of finding, usually to contrast it with another action. For example, it strongly implies something like: "I looked for it everywhere, but find it, I did not."

Why does the verb order change to heb ik instead of ik heb?

This happens because of the Dutch V2 (verb-second) rule. The finite (conjugated) verb must sit in the second position of a main clause. Since the first position is taken up by the fronted participle Gevonden, the verb heb takes the second spot, forcing the subject ik to invert and move to the third position.

Where does niet normally go, and why is it at the very end here?

In standard word order (Ik heb de fietspomp niet gevonden), niet is placed after the specific direct object (de fietspomp) and before the other verbs at the end of the clause. When gevonden is moved to the front for emphasis, the rest of the sentence structure stays the same, which leaves niet stranded at the very end.

Could I just say Ik heb de fietspomp niet gevonden instead?

Yes, and that is the standard, neutral way to state the fact. You should only use this fronted version (Gevonden heb ik...) if you specifically want to emphasize a contrast—perhaps someone asked you if you finally found the pump, and you want to stress that despite your efforts, finding it is exactly what you did not do.