How likely is it that a child will inherit diabetes and/or arthritis?

Uncertain : Type 2 diabetes has some genetic component as well as environmental. Bad lifestyle habits can increase the risk if one is genetically predisposed. Rheumatoid conditions like RA are not well genetically defined by familial patterns. So hard to say.

Depends: If one or both parents pass on genes that increase the risk of disease, this risk can be modified by diet and life style. A diet of unprocessed wholesome food, sufficient vit d3 during pregnancy and tihroughout life, avoidance of passive smoke, healthy bacterial flora in the GI tract promoted by eating plenty of vegetables, a good night sleep, movement , and stress reduction promote health.