fibroblast cells and fibers
Regarding dimensions:



As per this image the length and breadth seems to be ~30-50 μm (area should be roughly around 900 μm²). The third dimension (thickness) as per this article can be assumed to be around 3-7 μm. 

Regarding cell attachment:

Cells attach to extracellular matrix via integrins, which attach to ECM proteins like collagen and fibronectin. The matrix is a net of fibres and the cell is not specifically attached to a single fibre but rather "sits"(as you said) on the matrix [see the below image]. A single cell also simultaneously interacts with two different types of ECM proteins such as fibronectin and collagen. 



During migration they move along the matrix using lammelipodia (actin filament polymerization in the direction of motion).  

Fibroblast express ICAM1/VCAM1 (Inter/Vascular Cell Adhesion Molecules) under certain conditions such as inflammation and bind to T-cells and endothelial cells.