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Is there a way to differentiation between refreshes and actually first visits possibly with a header request?
Not specifically, but you can set a cookie or session variable.
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Is the date function accessing my server's date, the computer I edited originally from on, or the user accessing the sites computers date settings?
Of course the server's. PHP runs on the server; it doesn't have (direct) access to anything else.
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Will accessing a user's ip address with $_SERVER['REMOTE_ADDR'] ever come across to an anti-spam or virus program as a malicious site or present some question to the user?
No. PHP runs on the server; client applications never see it. The client's IP address is given out in every network transaction, or the server wouldn't know where to send the requested data.
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Not really relating to PHP but how does bandwidth work? I think I understand the basics of it. Like if you have 1000 users accessing 1byte of data at the same time you need 1+kbs of space free to have that function for all users. But the in a month feature I don't see how that works, unless they are just charging you by the chance that all your users hit you at the same time? Is that right or does it back up somewhere and the bandwidth can still be hit later by some of those same time users?
Bandwidth is not related to speed. It's the amount of data your users can download. If you have a 200GB/month bandwidth limit, your server can only serve a total of 200GB per month before being cut off.