No, at least not in the sense of 'invented before anyone else'. Trenches have been in use in one form or another for pretty much as long as people have wanted to defend somewhere. I'd put the ancestors of modern trenches in sieges of the Early Modern period, as soldiers are sheltering in the bottom of the trenches from enemy fire, though you could just as easily say that the distinction from, say Roman, trenches is too slight and they're essentially the same thing.

Trenches became more complex over time (picture of the French lines at the 1673 siege of Maastricht) as armies became larger and people tried to make siegecraft a more 'scientific' venture. By the time of the New Zealand Wars there were large sieges that turned into the kind of trench warfare that started to look a lot like what would happen in the First World War. The siege of Petersburg in the US civil war, which began about two months after the fall of Gate Pā, is sometimes cited as an instance of early 'trench warfare' (some maps and pictures on wikipedia), but so is the siege of Sebastopol in the Crimea (a decade after the Flagstaff war, about five years before the Taranaki campaign)

For your second question, pā developed a lot over time, based on the threats they faced. Pre-contact Māori had very few missile weapons and their pa were mostly made of ditches and banks, topped with a stockade. They were often constructed on hills or promontories, a particularly impressive example is One Tree Hill/Maungakiekie (picture for those of you not from Auckland). Some pa included fighting platforms, that the defenders could use to throw stones and use long spears from, placed above key points in the wall (an innovation that very obviously was developed in a society with limited missile capacity). For the second half of the question, as far as I know fortifications weren't used offensively in pre-contact New Zealand. As I said, there wasn't much in the way of missiles from the defenders to worry about, and Maori tribes usually lacked the logistical capacity for long sieges that might require defended camps (see for instance Belich, who talks a bit about the impact of the introduction of potatoes).

Maori fortification had to develop a lot after the introduction of the musket (there are some pretty good examples in the chapter on puhara I linked above). Pallisades tended to become lighter, and were supplemented by trenches behind them, and pa started to be built in more square shapes. Offensive earthworks were used in this period, as by Te Rauparaha (context for those that don't know: the guy who wrote this) at Kaiapoi (1831). A zigzag sap was driven up to the walls, and then brushwood was piled against them, and eventually fired. This is pretty much how a European commander would approach a similar situation, but as far as I know, Te Rauparaha was never in contact with Europeans that had studied military engineering, it was more of a case of similar problems leading to similar solutions.

During the New Zealand wars, while facing British artillery, further developments occurred, which resulted in the kind of fortifications that you would have been shown to support that claim (places like Gate Pā, or Titokowaru's Tauranga-ika). With their rifle-pits, trenches, partly buried houses and light palisades (pekerangi) that served a function similar to barbed wire, they were extremely strong. However, as I said above, American and European fortifications were probably of comparable strength. Offensive fortifications (except for the pa themselves, which were often quickly constructed and offensive in a strategic sense) weren't used against the British. Maori were considerably outnumbered and lacked artillery, so offensives usually consisted of guerrilla warfare.

I have rambled a bit, and put in a bit of unnecessary stuff, but my answers to the main questions are more or less: No, trenches have existed pretty much everywhere; Pa were used in intertribal warfare, but developed a lot in response to the different pressures they faced, and; yes, offensive fortifications were used, but only when they were appropriate, which was pretty much just during the musket wars.

Sources: James Belich's The New Zealand Wars and the Victorian Interpretation of Racial Conflict. (this is actually largely the source of a lot of the myths about Māori warfare, as it's a revisionist work that's trying to highlight their successes, but it's ok if you use it right) I also used 'Making Peoples' for something.

Ian Knight: Maori fortifications by Osprey Publishing (pls don't judge me, it's actually a pretty good book by Osprey standards)

J.W. Stack 'The Sacking of Kaiapohia'. The edition I have is in the back of The Stirring Times of Te Rauparaha by W.T.L. Travers (as you can tell from the title, it's extremely old)

The Pa Maori by Elsdon Best (also old, but the bits I've read are pretty good).

The various internet stuff I linked you to.