A lot of landlords won’t accept partial payments as it can create a mess about proving what payment goes towards what month.
I also wouldn’t suggest paying more than a deposit to hold the property before you’re scheduled to move in. There are scams out there and they could take your money and run!
You can open a separate bank account to put it in so you don’t accidentally spend it.
For your second question: that’s up to your landlord. Your landlord will likely want time to go in, deal with any damages and cleaning, do inspections, etc. And there maybe covid restrictions about how soon you can move in.
It’s possible you could sign something acknowledging the landlord wasn’t able to do cleaning/repairs and move in sooner. That will mean your lease will start sooner and you’ll still have to pay for that time.