Have you tried:-
1. Checking the 5V line at the chip leg/pin? (Is it 5V) and checking the 0V pin is grounded properly?
2. 'Scoping the 5V to make sure there isn't too much noise?
3. Putting a decouple cap directly across PICAXE +/- power pins?
4. Making a stripped down breadboard with just the recommended download circuit and THEN checking PWM?
5. Using the P.E. to check the PICAXE and firmware?
I find it difficult to check that complete breadboard, and my initial steps would be to do a 'checklist' of tests.
And certainly we can't tell the electrical integrity on the Forum... that's your job.
I get the impression that you have TOO MUCH in one go on that breadboard to do tests.
(Any bad connections, inappropriate connections, components, values can all affect operation. More components and connections = more variables in the circuit equation).
You should build up the project in stages.
e.g.
1. Get the basic download circuit working to programme PICAXE to flash a test res+LED from different legs.
Working? Yes:
2. Now download a drop of code to set PWM.
Check with res+LED or piezo sounder or preferably a 'scope.
Working? Yes.
then, 3. Add other things, one at a time, code-up and test....
or, No.
3. Check connections, code, PICAXE firmware,....everything.
Can't sort? Then swap out for another PICAXE.
Basically, if you've got the code right and downloaded OK, and it isn't working then there are three general possible faults.
1. The PICAXE is faulty.
2. The Power Supply is unsuitable.
3. Your circuit has a fault - and the Forum will rely on you to make the diagnostic tests.
Get the basics working before you use a DS1307, otherwise you can off at a tangent and waste time when the basics may be the root cause of the fault.
Tip 1 for newbies: Keep it Simple to start with.
Tip 2 for newbies:ALWAYS buy more than ONE of your chosen PICAXEs. In the unlikly event of a duffer you can swap out without waiting days/weeks for a replacment and saves carrige costs.
A duffer is VERY, VERY unlikely. The usual cause of failure is bad operation and ooops moments. This can happen to 'old lags' and Newbies so don't be shy!