For Buddhists

So if logic is unchanging, then is it eternal?
No. It is not unchanging, so it is not eternal. It is a human construct, which is why there are different logics invented by different people.

Side question, does eternal mean that it always will exist and also has always existed, or just will always exist?
For me 'eternal' includes past, present and future. I tend to use 'half-eternal' for anything with a beginning but no end.

Does Buddhism talk about how the illusion started?
Not to any great extent as far as I can see. Observably, ignorance exists here and now. Our task is to get rid of it, rather than to philosophise about its origins.

It seems like when the Buddha went on his path to enlightenment, he encountered many things. One after another conflicted with what he previously thought. These things led, eventually to him becoming enlightenment.
The Buddha tried many things. He lived in a Royal family, in great luxury. Having tried that he realised that he would not get what he wanted via that route. He then trained in yoga. That was useful, and gave him some of what he wanted, but it was not a complete answer. Then he tried extreme asceticism (as portrayed in a few Gandharan statues). That also was not the answer.

The failure of both material luxury and extreme asceticism, led him to try a Middle Way between the the two. That succeeded, and he reached enlightenment.

Does this mean that illusion can be a good thing?
Yes, providing it is approached correctly. The illusion of happiness from the material luxury of the Royal palace was necessary to show the (then) bodhisattva that material pleasures were not the correct path.

If so, do illusions have a purpose?
Not a 'purpose' as in some non-material thing attached to them. They can have a function, but only in relation to an observer, not anything inherent in the illusion itself.

Inevitably, there is a Zen story about this point:

On a cold winter night, a big snow storm hit the city and the temple where Dharma Master Dan Xia served as a Monk got snowed in. Cut off from outside traffic, the fuel delivery man could not get to the Zen Monastery. Soon it ran out of heating fuel after a few days and everybody was shivering in the cold. The monks could not even cook their meals.​
Dan Xia began to remove the wooden Buddha Statues from the display and put them into the fireplace.​
"What are you doing?" the monks were shocked to see that the holy Buddha Statues were being burnt inside the fire place. "You are burning our holy religious artefacts! You are insulting the Buddha!"​
"Are these statues alive and do they have any Buddha nature?" asked Master Dan Xia.​
"Of course not," replied the monks. "They are made of wood. They cannot have Buddha Nature."​
"OK. Then they are just pieces of firewood and therefore can be used as heating fuel," said Master Dan Xia. "Can you pass me another piece of firewood please? I need some warmth."​
The next day, the snow storm had gone and Dan Xia went into town and brought back some replacement Buddha Statues. After putting them on the displays, he began to kneel down and burn incense sticks to them.​
"Are you worshipping firewood?" asked the monks who were confused about what he was doing.​
"No. I am treating these statues as holy artefacts and am honouring the Buddha," replied Dan Xia.​

Those pieces of wood are neither inherently firewood, nor inherently Buddha statues. Statues and firewood are external to the wood, present in the minds of the beholders, not present in the wood itself. The wood is just wood.