In a healthcare setting, people are professionals, they are very careful and would never re-use a needle.
Also, to ease you up, you must know that it is practically impossible to acquire HIV though a used needle that was kept away from the syringe post usage on another individual. Since, the exposure of the contaminated blood with the environment would render the virus inactive. Nevertheless, the fact that it was used on a HIV positive person is debatable.
Rationally, you should accept the point that healthcare professionals wouldn't jeopardize your health, they are trained to take care of you.
While it is understandable that you are worried for your little one and you are concerned about it's well being, you must also realise that there will be some hormonal changes in these days to come, you might get anxious but try and not be worried much, your focus should be on good things, we wish you all the best.
Blood tests are painful because you're getting stuck with a needle. That is common sense. And the primary reason why sometimes blood is harder to draw up into the tube is because the patient is not properly hydrated. Ask anyone who draws blood on people on a regular basis and they will tell you that. Which is why I always drink 2-3 bottles of water before any blood draw to make sure they don't have problems getting enough blood.