Glad to hear that you have come so far. I have looked over the document that you have for the shipping and it seems very straight forward. The idea is that you pass along the values for the package to ship as URL parameters when contacting the service. The service will then respond with the rate or an error message.
I think the biggest thing for you to work out will be how you determine the package size to ship. If you make a request for each item then you could potentially have many quotes for the same order. If you make the request once for all items to ship you will need to calculate the dimensions of the package on the fly and use these in shipping quote request.
I think you should do this all with a single package. To make this easier after you add in your columns to the cart for the dimensions you should add in some calculated columns to hold the accumulated dimensions for that line item. This would multiply the particular dimension by the quantity so that you can calculate the entire dimensions for that line item.
Once you have done that you can just accumulate the dimensions based on the total of these calculated columns. You would then pass in the accumulated dimensions for all the items in the cart.
When it comes to sending the info to the server you should make use of cURL to do this, then you can capture the response to get the value for the quote. You can see an example of what cURL calls to servers look like in some of the eCart source files.
