Your boss is either having cramps, or is planning to fire you and is trying to build a paper trail to justify firing you.
I would guess it is the latter.
You need to document what happens so that when she fires you you can file a wrongful termination suit against the daycare and win a fat check.
Document things in a notebook - write down time / date / who was there. Not everything that happens, just the relevant things. Also - DON'T do it at work - do it at home and NEVER take your notebook to work, ever, for any reason.
You don't want to tell them that you are doing this - just do it.
When you get written up for something, make sure to add your comments to it before signing. Something brief like "Was advised by supervisor during training to turn in lesson plan on Friday" or "clocked in 2 minutes late because because no time cards were initially available" Don't argue with your supervisor when she writes you up - just add your comments to the form. Don't worry about the 1 minute late thing - any judge will see that as technically correct (on their part) but petty and indicative of their intent to wrongfully terminate you.
Since the handbook indicates friday for the "lesson plans" you are on solid ground there.
In the mean time watch your Ps and your Qs - and DO NOT be late ever again. Early is cool, late is not. Don't blow your fat check on hitting the snooze button.
When they do fire you, don't make a fuss, don't tell them you are going to sue them, don't tell them that you have been keeping records of what has been going on, don't threaten them. Just be gracious, thank them for the time you worked there and leave quietly and with your dignity.
If I were you I would start consulting with a lawyer now.