Re: Questions for a difficult yard
This might sound like a broken record, but you will never go wrong by adding manure or compost to your soil to improve nutrient levels and bio-activity. Organic matter will break up clay particles, leaving behind fertile, tillable soil.
I wouldn't worry about the small amount of urine where the dog has been over the winter, however you may want to consider a permanent "winter" residence/kennel area for the dog to prevent damaging grass through the winter months as well as tracking the muck into the house.
For your grass area I would recommend applying 3" to 4" of municipal compost or manure and work it 4" - 6" into the soil. For the garden area I recommend manure or organic compost not municipal compost because municipal compost isn't free of chemicals and other nastiness you don't want in your veggie garden. I would also say that two applications of manure or organic compost, worked to a depth of 6" to 12" would be best.
I suffer from CDO ... Its like OCD, but in alphabetical order, LIKE IT SHOULD BE!!!