Particle on XY plane (Not Z)?

Although I am not 100% sure which things you are envisioning going on which axes, one way to render a sort of "nest" using a strand object is to create curves (e.g. bezier curves) that define where the strands will lie. Then create a mesh object that is a cross-section of the strand and give it your material. Make sure to apply location and rotation (but not scale) to your curve objects. Then, starting with the first curve:

1. create a duplicate of your cross-section object.
2. give it an Array modifier, check "Merge", Relative Offset X:1, Y:0, Z:0, fit type: Fit Curve, and specify the first curve object.
3. then give it a Curve modifier, again specifying the first curve object.

If all has gone according to plan, your curve is now "stroked" with your cross-section object. Then repeat for each additional curve in your nest.

Nest with Vertical Strands:

Nest with Horizontal Strands:

It will be easier if you duplicate the last cross-section mesh you modified, because it will already have the modifiers applied, you just need to update the curve objects the modifiers are using.

The nice thing about this is you can modify your curves to create a "weave" effect, which can be considerably handier than trying to smoothly deform the meshes.

When things are to your liking, you can apply the modifiers (and remove doubles, there will be a LOT of doubles) and the resulting mesh objects can be further modified . . . for example you can join the strands into a single weave object deform it along a curve with another Curve modifier, which might come in handy in making something like the image you provided.

A different technique would be to simply bevel your curves with a curve that represents the cross-section of your strand.

Select object A, then add a Wireframe modifier.
Increase thickness to desired width.

What I gather from your description, you are trying to accomplish this:

First, move the origin of A to an edge.

Num-7 for top-view.
Click on the leftmost edge of A
Shift-S snap Cursor to Grid
Object > Transform > Origin to 3D Cursor

Next, set up particles on B.