I realise that this is an old post, but this info may be useful to someone.
I can confirm that the time and date can be set programmatically via i2c, and therefore possibly by a web page as well, though I could not do it using the provided page which states that it works on the shared memory bytes and therefore may filter the input (I haven't studied the coding). It is not necessary to use the Wizard or the manual configuration mode. You could always pass a value from a web page to the picaxe and back to the DS1307 if you can't write directly, but I can't see any reason why your own web page would not work.
You will need the DS1307 data sheet if you want to change the control bits or to code BCD correctly for the device, but you can download that from MAXIM and it isn't difficult to use.
Byte 3 is the day of week, which can be used even if not provided for by Picaxe.
The PNS does not seem to use the time so probably won't be bothered if you change it. The manual is therefore a little over-cautious.