Basically, none of the things your ped said sound like things a three year old should definitely know, in my (non-professional) opinion.
I would not expect a three year old to be able to write any letters at their third birthday. As they approach their fourth birthday, a child who had been in preschool should probably be able to write their name, but even then, a child who can't isn't really behind. Writing your name is a skill that children should definitely have upon entering kindergarten; before that is good but not something they can ALL do.
Most kids turning 3 also can't draw a picture of mommy. At that age, a picture of a person will probably be a head with arms and/or legs coming off of it, but without a body. There may or may not be facila features.
A girl turning 3 should certainly be ready to potty train if you haven't done it already. Many girls can potty train earlier, like 2.5, but if they haven't done it by 3, it is definitely time to start. Pull ups at naptime and overnight are still very common, even at age 4 and sometimes 5.
As for the address and phone number, very few 3 year olds will know this. Many, but not all, will know this entering kindergarten. It is one of the assessment questions that kids entering kinder are asked in my city.
Go online and search for the "ages and stages questionnaire." It is a really wonderful questionnaire that covers communication, fine and gross motor skills, problem solving and more. They have different versions for different ages, so you can find one for age 36 months and see how your daughter is doing compared with basic expectations for that age. I don't have one for that age, but here are a few questions from the 33 month one:
Communication
1. Can your child identify 7 body parts (ex: point to your eyes, ears, etc)
2. Does your child speak in sentences that are three or four words long?
3. Ask your child to do one simple task (ex: put the book on the table) Does she carry out the directions correctly?
4. When looking at a book, does your child tell you what is happening (for example: barking, running, eating, or crying)? You may ask "what is the dog doing?"
5. When you ask "what is your name?" does your child say his first name or nickname?
Fine motor:
1. After your child watches you draw a line from the top of the paper to the bottom, ask her to make a line like yours. Do not let her trace yours. Does your child make a single line in a vertical direction? (it can be curvy/wiggly, doesn't have to be perfectly straight as long as it is generally vertical). Can they do the same for a horizontal line? A circle? (the circle doesn't have to close perfectly or be exactly round, but should be one major penstroke, vs a tornado-like squiggle or only a "c")
2. does your child turn pages in a book, one at a time?
Problem solving
1. when looking in the mirror, ask "Where is (child's name)?" does she point to her image in the mirror?
2. Have her watch you line up for objects, like blocks or cars, in a row. Does she copy or imitate you?
3. When you say "Say 'seven three,' " does your child repeat just the two numbers in the same order?
4. After your child draws a picture, even if it's just a scribble, does she tell you what she drew? say "tell me about your picture" to prompt her
Personal-social
1. Ask your child "are you a girl or a boy." does she answer correctly
2. Does your child use a spoon to feed herself with little spilling?
3. Does your child put on a coat, jacket or shirt by herself?
Anyway, I know this is a long answer, but I hope it helps. If you answered yes to most of those questions, you have nothing to worry about. If you didn't, you may want to talk to a professional, though from your post it sounds like your daughter is pretty much on target for her age.