If I pay a bill online will the money leave my account straight away?

Not always.  It can take days (or longer).  Don't buy something without checking first.  You should be able to look at your account history and see if the money has left it or not.

some debit you immediately, some don't ..............................

Some online retailers don't debit your account until the item ships. Others do it as soon as the order is placed.

Yes, no, an maybe.

Did you authorize payment through your bank, or through the website of whoever you are paying?Is the company set up to receive payments from your bank electronically, or does the bank actually have to mail a paper check?

The REAL answer is check the transaction history on your bank account an see if the payment has posted.

That's a little confusing, but based on your comments below it sounds like you're wondering what your current balance represents.  Does it include the bill pay transaction or not?

The simple answer is that you need to look at your transactions to see.  You can do that on the website, on the phone, from an ATM, or even in person.  

bills are not paid without money first.

yes it will, but it will not show up on your statement for 3 or more days

 "but I don't want to leave myself short"

Then don't spend more money than what your register balance is.  In other words, assume all money you spend leaves your account immediately and act accordingly.

ETA:  If you're trying to say you have adequate funds for both purchases, then you're right, I have no clue what your question actually is.    How can you be short if you have adequate funds?