Ingress: Do new fields within old fields form if the old field expires?

No. New fields cannot form on their own.

Let's assume I've linked the neutral portals

However, consider the following situation:

This will take these steps to reproduce:

• build field 1
• with two additional keys, build field 2
• with one additional key, build field 3; at the same time, the larger field also appears, giving you 313 AP for the one link, 1250 AP for the small field 3, plus another 1250 AP for the larger field

In my drawing, the third picture shows the double-layer with a darker shade of blue. Now if the anchor in the middle gets destroyed, the large outside field will still be standing.

If however one of the outside anchors would be destroyed, three fields would disappear, and it would look the same as the first, top left illustration.

No, fields are only formed at the instant when the third link forms a closed triangle (or triangles, as the case may be for a double-field).

As a sidenote, you can capture the portals inside the field and link from the edges to form fields.

No because in the first place you can not link those portals inside among themselves because you can not link portals that are inside fields.

What you can do is capture those portals and create links FROM the outside portals to them. If that creates fields, fine. If not, then no new fields will be created when the outside portals fall either.