Firstly, your family $\mathcal{F}$ is not closed under conjugation if $H$ is not normal. Depending on what you want to do, this may not be an issue.

There are two references for the construction of $E\mathcal{F}$ that I know of (but I would be glad to hear of more). The first is tom Dieck's book *Transformation groups*, where in Chapter I.6 he states that a model for $E\mathcal{F}$ is given by the infinite join
$$
E\mathcal{F} = X \ast X \ast \cdots
$$
where $X=\bigsqcup G/H_a$ is a disjoint union of orbit types. This is a $G$-CW complex in an obvious way, and suggests that the answer to your second question is no.

The second reference is Lück's *Transformation groups and algebraic $K$-theory*, available from the author's webpage (scroll down to Books). Apparently, using the results in Chapter 2 of that book, one can build a $G$-CW complex model for $E\mathcal{F}$ by an iterative process of attaching $G$-cells. The construction is not carried out explicitly, however, and I'm not sure of the details (perhaps this is the topic for a separate MO question).