so first off, you seem to be in need of more support yourself. I get the feeling from reading between the lines that you are getting close to being pushed past your limit of what you can tolerate. It is a good time to seek more help from your spouse, relatives and a psychotherapist if you can find one. You must take care of your self or you can not take good care of your child!
Next, I am going to give you the good news that much of what your daughter is doing sounds like typical child development at this age. I know that does not necessarily make you feel better, since lots of what young children naturally do can be annoying, inconvenient, or hurt our feelings. Here are the issues you raised that reflect typical early childhood thinking:
1. Not being able to answer 'why' questions, particularly when confronted about negative behavior. She can not answer 'why' questions well, and is probably so anxious about having displeased you that she can not have a conversation about what she has done that made you angry. Confrontation makes little children confused and distressed--best just to tell her firmly what exactly you want her to do next time and give lots of reminders and praise when she makes a good choice.
2. Potty--full control is not expected until around age 6. Expect that accidents will happen, and that they are truly accidents.
3. Her sense of time--essentially her sense of time spans only a few minutes. She has very limited awareness of how much time as elapsed, how long any given time period actually is, and how to be patient when she wants something. She also can not reliably predict cause and effect in the future.
4. Empathy with you and the dog--children are capable of empathy, but only for very concrete situations like if they see another child fall down and cry. She is way too little to be able to take your perspective, thus she can not tell much about why you may feel grumpy, angry, or hurt because of her actions. Your responses probably surprise her. You will need to label your feelings for her, and teach her what she can do to help (e.g. You kicked the dog and hurt him. I want you to pet him with gentle hands to help him feel better).
5. Memory--even bright children have very limited memory at this stage. They may remember random things from long ago, such as what color the cake was at their birthday party, but do not expect them to reliably recall information on command. They are easily distracted and genuinely forget directions without frequent prompts and reminders. Not only that, but they are often not fully attending to what you say, and so may miss much of your directions (particularly if you give complex directions or multistep directions).
If you want to learn more, check out these books: Your Child's Growing Mind by Jane Healy, How to Talk so Kids will Listen and Listen so Kids will Talk, and The Bilingual Edge by Allison Mackey.
Finally, I am sorry but I do not know enough about the health system in Canada to advise you in finding a pediatrican.
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