Is Heimerdinger able to use all tap abilities at once?

No, Heimerdinger, Inventor cannot use all of his copied tap (exhaust) abilities at once.

According to the official rulings for Heimerdinger:

• You choose only one: Even though Heimerdinger gains all exhaust abilities from friendly legends, units, and gear, you must choose exactly one of those abilities to activate when you exhaust him.
• One activation per exhaustion: Activating any of these abilities requires exhausting Heimerdinger as the cost. Since you can only exhaust him once per turn (unless he is readied by another effect), you can only pay the cost for one ability at a time.
• Not simultaneous: The abilities do not trigger or resolve simultaneously; they are distinct options, and you select one to use each time you pay the cost.

In short, while Heimerdinger has access to all those abilities, he functions as a single source that must be exhausted to trigger one specific effect.