Since rancid oil produces free radicals, would that rancid oil if mixed with fresh oil cause the fresh oil to go rancid as well?
Short answer: No.
Long answer: No - because the free radicals are very short lived and highly reactive - each free radical will generally only interact with one other molecule before being destroyed. This could cause degradation of reactive nutrients such as vitamins and other nutrients in the oil, but is unlikely to cause much problem with other foods (I would guess). They also have a very limited interaction with the food which the oil is covering (surface area only), so they have a limited chance of causing damage. You are much much more likely to have bacterial or fungal degradation of the food than free radical.
I would have thought that you are more likely to throw the other food out because it is coated in unpleasant tasting oil rather than be worried about the degradation of the food.
Note that rancidification is the production of smelly compounds, mostly aldehydes, from the long-chain fatty acids found in fats and oils. This is a property peculiar to fats and oils and is generally not a problem for other food types as they do not contain large amounts of the fatty acids, so can't get "rancid" as such, though they can be considered spoiled if the fats within them (e.g. fat in meat) go rancid.
I doubt about the needing of an answer as it is self-contained in the question but it seems worth a chemistry clarification.
Oxidative rancidification goes indeed via radicalic pathways. In "fresh" oil, the process will be eased by, e.g., peroxides already formed in the already rancid oil that you would eventually mix.
Peroxides are relatively unstable and the homolitic breaking of their O-O bonds leads to two radicals that re-initiate the process.
This is how the chemistry goes.
However I can't judge how faster will be the rancidification of the fresh oil as compared to that of the same let alone. There should be a difference but perhaps indecteable from an organoleptic stand point. Except for the following...
Beside the fact that new rancid products form or not, a small amount of deteriorated rancid oil can certainly ruin the fresh oil depending on how it will be used in the kitchen. Some mixing could be done to fry probably but should be strictly avoided to season a salad, isn't? This is the only part in which I do agree with another answer.