Urine from healthy dogs (and humans) is sterile on leaving the body. It can, however, encourage the growth of existing bacteria: so these you would have been exposed to anyway.
Faeces are not sterile at all and fresh faeces from pretty much any animal can be very harmful to humans. Dogs frequently carry parasites which are easily transmitted to humans and can cause a variety of problems - not just on eating but transfer from unwashed hands to eyes can lead to serious eye conditions that may result in blindness (this is why many parks hand out very heavy fines to dog owners that foul these areas).
It is best not to grow salad or root crops in areas that have been fouled, however, stuff that it is harvested away from the splash zone (both rain and urine!) should be safe. Fruit from fruit trees would be unaffected by a dog urinating on the trunk or fouling the root zone.
You initially state lettuce (Lactuca sativa), but then mention perpetual spinach (not actually a spinach at all but a beet Beta vulgaris subsp. vulgaris). These are not remotely related. Whilst hazards are the same regardless of the species, it does have implications for what you can do with them: lettuce is rarely cooked whereas perpetual spinach is. If you were to cook the plants then this would remove the biological hazard. I would advise against eating any uncooked food that has come into contact with raw faecal matter. Thorough washing may not be sufficient to prevent illness.
As for 'chemicals' found in an unnatural diet the level of exposure in this instance is insignificant. Having worked as chemist I cannot help but chuckle when people get worried about trace contaminants in our environment given the far more serious risks they are prepared to expose themselves to (alcohol, smoking, crossing the road etc.).