Your post covers a wide range of both personal and professional concerns. The root of these "mistakes" you describe seems to be rushing and distractions. Grabbing the glucometer docking station and putting on the blood pressure cuff the wrong way are the product of trying to perform an unfamiliar skill too quickly. Anxiety about the skill and feeling pressure to perform it well are also probably contributing to these missteps. The best way to combat this is to simply slow down. Take breath and think through your actions before your start doing. Walk through it step-by-step before you begin a skill. Run through the "what-if" scenarios in your head:
"What if I'm doing this dressing change and my sterile glove touches my scrubs
?... I'll stop, explain what happened, take off my gloves, wash my hands, and put on a new pair... I'd better being an extra pair of sterile gloves with me just in case."
By doing this, you gain confidence that you know the steps, you've thought through ways in which it may do wrong, and you've prepared yourself by identifying what extras you might need to bring with you. If you're in the middle of a skill and something seems off, pause. Look around you, think about what you've done to that point and trouble shoot what may be going wrong. Trust me, you will look far more competent if you stop and troubleshoot than if you continue fumbling to correct the issue without taking time to identify what the issue is.
The driving mistakes you mention seem to come down to simply not paying attention. Which, despite your perfect driving record, is dangerous and concerning. Where I am from, bus drivers will call take your plate # and call the police if you pass a stopped school bus. Make a special effort to be focused and present while driving. Reduce or eliminate distractions like the radio, cell phone, and conversations with passengers. You might consider a defensive driving course as well.
The social issues you mention... I can't really help you there. You seem to be pretty critical of yourself and how you come across socially. My biggest piece of advice is to embrace who you are. Some of this will come with age. Maybe you're the energetic, outgoing person who loves singing karaoke. Maybe you'd never be caught dead singing in public. Maybe you're the somewhat dorky person who loves Star Wars and science. Maybe you're not great at making small talk, but give really great advice. None of these people are better than another. You don't need to act a certain way to make people like you. Your friends should like you for who you are, but you need to like you for who you are first. When you realize that your true personality is awesome, and embrace it instead of evaluating it, you'll be less anxious and happier.