I haven't seen any specs on pushing capacity for loaders. Mostly they loose traction and spin 1,2,3 or 4 tires - that's when you are pushing too much. As to pulling, standard FEL operations include filling the bucket to max capacity, putting it in float and back dragging. You can also add down force and back drag to help level and compact the soils.
I wouldn't be at all hesitant to push or pull any log that the loader could lift. Always be cautious with a log when pushed as it is able to swing around and smack you (or the equipment). This always happens with abrupt speed due to catching a stump or rock in the exact perfectly wrong way.
They make skidding cones that fit over the front of logs and there are angled skids that the front of the log can be chained to. These help the logs bounce over stumps etc when pulled.
As far as damage - that is usually from shock loads. Whacking into objects that don't move. Also unbalanced loads. If you are using forks on the bucket to lift a log and it rolls back and down hard, the cylinders can bend. ( Got a buddy with a banana cylinder from that exact thing).
jb