Are pesticides used on fruit easily washed off?
Pesticides never were my main field of research, so you might want to take the following with a grain of salt:

Most pesticides i'm aware of are insoluble in water, but you might want to cross-check that for particular substances in the huge Pesticide database of the EC.

This insolubility seems reasonable, otherwise the substances would get washed off after the next rain.

Concerning your second question, the answer unfortunately is yes. To my knowledge, different pesticides are typically applied at the various state of plant growth. In order to suppress weeds, one might spray fields in an early, pre-emergent state. Later, fungicides or other substances might be applied during blossom. In order to protect the bees, it is however not recommended to apply insecticides during that time. Reasonable and minimized use of pesticides will keep the load in the fruit close to the limits of detection or beyond. Depending on the origin of the fruit, this might however not be the norm. In these cases, high concentrations of pesticides, even those banned in your home country, are detected both in the fruit and its products. 

This leaves us with the question what we can do as consumers. If the fruit is highly contaminated, we can't do anything. Therefore, it might be reasonable to avoid products from countries infamous for breaking environment protection rules and go "organic" instead. Honestly, i never peel apples, but i wipe the surface with a piece of cloth. But i avoid industrial sorts, buy from local farmers, or pick them from the ground of meadow orchards.

I hope that this contribution, although more personal and less scientific, was helpful.