What exactly is "price today"? Is it the exact current price of the stock?
It depends on your definition of the indicator. You can choose "close", "open", and several other options.
Let's consider "close" only.
If you set the Period to be a Day, then the most recent data point is either the current price or the price of the previous day's close.
Why is it that when looking at EMAs in charts on the web, they can change every minute even though EMAs are daily?
EMAs are not daily; the Period is set by the user and can be 1min, 1hr, or 1 Day, along with several other alternatives. The requirement is that each discrete value of price data represents the same amount of time, but the charting software that you use does this automatically for you once you set the Period.
Are EMAs only for days?
No, see previous answer.
When people talk about a "200 EMA", do they mean N=200? A period of 200 days?
Yes, it means N=200, but the units (Period) are set by the user and can be days, though they don't have to be. N is known as the Length and is unitless on its own before pairing it with a Period.
If the EMA today is based on the EMA yesterday, how do we find the first EMA?
You use the SMA for the first N data points, and then use EMA for all of the following ones.
This is an example: