How can an infinite universe expand?

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Nope. Space is expanding. That is what so-called Dark Energy is all about.

It would probably surprise people to know that every closed segment of the real line has precisely the same number of points as any other closed line segment regardless of the length of the two line segments.

For example, [0,1] has the same number of points as [0,2]

Define f:[0,1]->[0,2] by f(x)=2x.

Then f maps every point in [0,1] to a point in [0,2] and so [0,2] has at least as many points as [0,1].

Similarly, define g:[0,2]->[0,1] by f(x)=x/2.

Then g maps every point in [0,2] to a point in [0,1] and thus [0,1] has at least as many points as [0,2].

Therefore, since there are as many points in [0,1] as [0,2] and as many points in [0,2] as in [0,1], both line segments, [0,1] and [0,2], have the same number of points.

There is no such thing as a void. Space and time is all there is. It cannot 'expand' into the void. If we represent dark energy in the Friedmann equations (as lambda, the cosmological constant) you can see it is causing an accelerated expansion of space, via minimizing the critical density ratio (p/pc). Thus, matter is become less and less commonplace, and is instead replaced by an expanding vacuum.

If we lived in an infinite universe, then the heat death would have happened infinitely long ago.

Yes it can. There are different magnitudes of infinite. Look of Cantor Set Theory.

For example there are more real numbers than there are integers.

Not necessarily.

A finite sized universe reaches heat death in finite length of time.

An infinite sized universe would take an infinite amount of time to reach entropy. So it balances out.

Nope.