I'm not sure you've appreciated the magnitude of what you are asking for. It is along the lines of, "I know what an atom is; how do I split one ?"
The real problem is that unless someone has done what you want using the hardware you've chosen then there isn't a readily available solution or answer; at best only theory and ideas and you'll have to do some experimentation and design work.
As you want to drive the ultrasound TX directly from the PICAXE you need something to provide those pulses. A single pulse can be generated by PULSOUT, a burst of pulses by using PWM or by using PWMOUT. Having chosen the 40X you have the choice of using PULSOUT or PWMOUT.
It may be possible to use multiple PULSOUT's to generate a stream of pulses, but because of the speed of the PICAXE, there will be around a 250uS delay between them, and you could then be handling PULSOUT's when the echoed signal is returned.
To generate a burst using PWMOUT means setting it up to send the right frequency, then turning it off immediately afterwards, and again hoping that is all done before the echo is sent back.
To determine the range requires measuring the time between the pulse being sent and the echo received. As shown in the BS1 example, this can be done by using PULSIN with appropriate hardware.
This time is then converted to a range as described earlier - distance=speed/time. Because the PICAXE only uses integer maths, and the 'time' needs calibrating then this will require some fudging, along the lines of ...
- distance = time - K * M / N
The values used will depend upon the units of distance you want (cm), what values of time you get, and the equation could get complex in avoiding overflow. To start with, don't worry about distance, just concentrate on getting a time back which increases as the object reflecting is moved further away.
The PICmicro code isn't really relevant to a PICAXE solution because it operates in a time domain much smaller than the PICAXE can, and uses techniques not applicable to the PICAXE. The hardware for the PICmicro uses RA0 and RA1 to drive the ultrasound TX by using out of phase on each pin, which can't be done on the PICAXE. Unless you use and external driver, one leg will need to be tied to 0V and the other driven by the PICAXE. If using PWMOUT the pin used is predetermined by the pins which PWMOUT can use.
The first step is to build a PICAXE transmitter and the receiving hardware so you can check that ultrasound does pass between the two and not when blocked. That proves the hardware.
Then you'll have to experiment to try and make what you have work as you want. You'll probably need access to a scope to see what's going on. To use PULSIN as per the BS1 example code, you'll need a receiver which gives a waveform which can be used with PULSIN. If the circuit you've chosen doesn't work that way then I'm not sure how you will go about reading the signal and determining the time for the echo. If you don't know what signal the receiver gives in response to a TX burst, you aren't likely to get off the starting blocks.
If you are looking for a ready built solution which has all the answers to your questions, we are back to the suggestion that you use a ready built module which has example code for uses ( the SRF04 interface software is in the PICAXE 'samples' directory ), or something like the BS1 example which should work as described on a PICAXE. Porting a solution from a differenct device to a PICAXE is not as straight forward as may be imagined and relies upon having a depp understanding of how that solution works and ensuring it will work with a PICAXE. Unfortunately you are asking for the solution to a problem which no one on this forum has probably set about solving.
Perhaps someone who has used a PICAXE to do range finding other than using an SRF module will come up with a answer along the lines of, "I did this ...", but otherwise you will be pretty much on your own.
One such example I've found is here ...
http://www.siliconchip.com.au/cms/A_103243/article.html