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> Universe is infinite. It means, that we can zoom up infinitely AND we can zoom down infinitely, which means that space always filled with something, and this something have a temperature.

I'm not sure this follows. The set of positive integers is infinite. It means that there's always a larger integer, but it doesn't mean there's always a smaller one, or an infinite number of integers between 1 and 10.

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Even in this case, there is infinitely small chance that we are sitting so close to the beginning of everything.

This is only true if we pick a scale with a fair infinite-sided dice.

If it is as we suppose that the universe started all squeezed into a point at the beginning of time, and then explosively expanding out from there, then I don't see why we would expect a uniform distribution like that.

Replace integers with rationals or reals.

Having four legs doesn't follow from something being an animal just because there are examples of animals with four legs. A chicken is also an animal.

Having four legs is accidental to being an animal, just as there being an infinite number of elements between any two elements in a set is accidental to its cardinality being infinite.

Chickens have wings at the end of their arms.

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