Sorry to hear you're feeling scared - don't worry it's normal, not silly ! Even if you're not usually an anxious person, pregnancy hormones can make you feel anxious at times and it's normal to feel protective over the baby and worried that something might be wrong - it's mothers instinct already kicking in.
I also used to be so scared of needles, always fainted and hated the idea of having bloods done but actually I had an amazing nurse take my blood it was so so quick I hardly even noticed she'd done it ! I just told her I was a bit nervous about needles and she was so quick. Also you'll get more used to having blood taken it becomes less of a big deal the more you do it - those nurses take blood all day and are complete pros at it.
At the first midwife appointment they just took a urine sample and blood sample, took my blood pressure I think, weighed me. Asked me and my partner questions like when was last period, have I been taking folic acid, did I/do I smoke, drink etc. How have I been feeling so far... Those kind of basic questions. They also ask you to fill out a lot of these questions in a form in the waiting room eg - if you've had STD's or bladder infections or any other medical problems in your history. Then they discuss your answers with you in the appt. They ask you random official questions like If you would consent to a blood transfusion (random but guess they have to get your permission as Jehovah's Witnesses don't believe in blood transfusions?). It was very chilled and I felt like we had a lot of time and attention in that first appointment. Subsequent appt when I went on my own to discuss my blood results/have general check up was much quicker. They don't even touch your stomach or do an internal examination at first at all so don't worry about that. They'll probably book you in for a scan at this appt too - probably on a different day nearer to 12 weeks. They may send you to Ultrasound department so you can book this yourself after your midwife appt. The midwives are very patient and caring in my experience so don't worry about them 'losing patience' - they are there to reassure you and answer any questions so if you've been feeling a bit crappy and anxious - when they ask you how you've been feeling just be honest ! I'm sure they'll reassure you. Being overly emotional is totally normal in first trimester - as you get more info about your baby and have more scans and get more used to the concept of being pregnant, you will start to feel more comfortable and calmer.
Don't beat yourself up about your feelings - just let them come and go and give yourself permission to feel the feelings and then let them go when they go. You can feel a bit like a zombie/a different person in 1st trimester for sure - I did at times ! Felt like my whole body/mind had been taken over - which is kind of has - your body is very busy building a life so be kind to yourself and just take everything as it comes - you're allowed to feel emotional and anxious, it's ok. But first midwives appt is really chilled and straight forward so if anything try and look forward to being one step closer to getting your first scan booked, visiting the hospital where you'll be giving birth - see it as a step on the journey to meet your baby and a chance to ask questions/have someone ask you questions about your pregnancy and how you're feeling - which is great !
And most likely everything will be fine with baby and with you ! Good luck and hope you feel better soon.