Impossible to pack Circles without gaps

It is intuitively apparent that circles cannot be packed without any gaps. I thought this is easy to prove, but it turns out not to me.

I have $2$ versions for this question, which likely to have opposite answers.

$1:$ Is it possible to pack finitely many circles(of radius larger than 0) in the same size within a finite region.

$2:$ is it possible to pack circles(of radius larger than 0) within a finite region. (Which means we can shrink the size of the circle as small as we want and there can be infinitely many of them).

For $1$, I thought it is obviously impossible, since no matter how we arrange the circles, there is always some rooms not included within the circles. I thought it is easy to prove until I realise that there can be more way than I thought to arrange the circles. (see the pictures: or maybe this is already a proof?)

For $2$, I think this is possible, just like pack any shape by rectangles like Riemann Integral, But I have not came up with a proof.

I think these are not obvious questions and need some tools, which geometrists may have but I do not. Any ideas and suggestions will be appreciated.