Every phrase highlighted is discussing their capacity to survive in situations of trouble.
* "If it came to it", where **it** is _a situation of trouble_.
* "when put to it", where **it** is overcoming _a situation of trouble_.
* "survive rough handling by grief, foe, or weather", where **rough handing** is _a situation of trouble_.
The text is explaining the counterintuitive capabilities of the characterised people. Because of their appearances people would, intuitively, consider hobbits to be incapable of surviving well.
* Grief is a situation where they are losing a loved one.
* Foe is an opposer that intends to harm them.
* Weather is an act of nature that delivers a problem.
Being rough handled by any of these situations means the hobbits are suffering.