Doctors note when sick?

I was sick last week and called in for one of my 3.5 hour shifts (with plenty of notice). I got worse and went to my doctor yesterday, and found out I have pneumonia. I work tomorrow and Friday for the same length as well, and called in today to let them know that for sure tomorrow I won't be able to make it.

They said that's fine and then at the end asked me to bring my doctors note when I come in on Friday (I said I wasn't 100% sure if I'd be well enough on Friday but that I hope so). My doctor never gave me a note and she's only in on Tuesdays... I'm part-time as I'm in high school. Does it have to be from my family doctor? Or could I go to a walk in and have them do it? They usually charge fees for them too. I'm new to retail - is it standard to have to have a doctors note?

I'm also terrified of being fired since I only started in October. This is a good paying job and I quite enjoy it, plus depend on it for my car insurance and savings. I've now had to call in sick twice, and they've made errors on my time off (which was approved) and I've had to talk to HR several times. I've never worked for a big company and I'm just very intimidated by it all and how easily replaceable I am, especially since I haven't even been there 3 months.

Edit: I know it's totally within theirs rights and if asked I must supply one. I'm just concerned about if it's not my family doctor, and if this is standard or are they suspicious of my time-off. I'm really scared and hate this whole situation. Wishing I weren't as sick as I was so I could work but really can't help the fact that I'm in bed with pneumonia...