You might want to consider registering your copyright. Having a copyright registration certificate that pre-dates any copies can be handy. I've had server hosts require this as part of their take-down procedure.
Regarding archive.org, you might want to submit your site for indexing in the Wayback Machine, so you'll have a record (starting with a spidering in the next few weeks, with your listing appearing probably within a year after that) of your site as it exists now.
But the Wayback Machine would not, to my knowledge, be acceptable evidence in a court case (regarding, for instance, a plagiarism claim). At best, it can be helpful in convincing some that you published the content online first. But opponents can still claim that they published the content in, say, book form long before you put the content on the web, or claim that they published on the web before you ever did, but were never indexed by archive.org.
In other words, such an online record, while sometimes helpful, is not to be relied upon when you "need" to "prove" something.
And giving a lawyer a CD only "proves" that you burned the content onto a CD on such-and-such a date (assuming nobody points out that you can change the clock on your computer). It doesn't prove that you own said content.
Registration is not "required" to prove ownership, but registration (with the appropriate legal authority) is accepted in court as proof of this ownership, and may greatly ease the legal burden of the content creator/owner, while possibly increasing the judgement awards handsomely.
Registration is not to be sniffed at.
This is called "the poor man's copyright", and is an urban myth. A sealed envelope is not regarded as legal "proof" of anything.
(Sample argument of opponent: "Plaintiff could have mailed himself an empty unsealed envelope two years ago. Now he needs 'proof' of ownership, so he back-dated a copy of his web site by tinkering with his computer's clock, burned the CD, and sealed it in that already-post-marked envelope. Your Honor, this envelope proves nothing! Move for dismissal and costs.")
If you want "proof" that will be accepted in court (or that will at least create a much-greater burden on the other claimant), file the paperwork, burn the CD, cut the check, and register your copyright.
I'm not a lawyer, but I've been down this road before. For specifics, consult with a copyright attorney in your country, and/or review the documentation for your country's copyright office.