Python program on server - control via browser
Django is a good candidate for the website, however:

It is not a good idea to run heavy functionality from a website. it should happen in a separate process.
All functions should be asynchronous, I.E. You should never wait for something to complete. 

I would personally recommend writing a separate process with a message queue and the website would only ask that process for statuses and always display a result immediatly to the user

You can use ajax so that the browser will always have the latest result.

ZeroMQ or Celery are useful for implementing the functionality.
You can implement functionality in C pretty easily. I recomment however that you write that functionality as pure c with a SWIG wrapper rather that writing it as an extension module for python. That way the functionality will be portable and not dependent on the python website.
I have to setup a program which reads in some parameters from a widget/gui, calculates some stuff based on database values and the input, and finally sends some ascii files via ftp to remote servers. 
In general, I would suggest a python program to do the tasks. Write a Qt widget as a gui (interactively changing views, putting numbers into tables, setting up check boxes, switching between various layers - never done something as complex in python, but some experience in IDL with event handling etc), set up data classes that have unctions, both to create the ascii files with the given convention, and to send the files via ftp to some remote server.
However, since my company is a bunch of Windows users, each sitting at their personal desktop, installing python and all necessary libraries on each individual machine would be a pain in the ass.
In addition, in a future version the program is supposed to become smart and do some optimization 24/7. Therefore, it makes sense to put it to a server. As I personally rather use Linux, the server is already set up using Ubuntu server. 
The idea is now to run my application on the server. But how can the users access and control the program?
The easiest way for everybody to access something like a common control panel would be a browser I guess. I have to make sure only one person at a time is sending signals to the same units at a time, but that should be doable via flags in the database. 
After some google-ing, next to QtWebKit, django seems to the first choice for such a task. But...
Can I run a full fledged python program underneath my web application? Is django the right tool to do so?
As mentioned previously, in the (intermediate) future ( ~1 year), we might have to implement some computational expensive tasks. Is it then also possible to utilize C as it is within normal python?
Another question I have is on the development. In order to become productive, we have to advance in small steps. Can I first create regular python classes, which later on can be imported to my web application? (Same question applies for widgets / QT?)
Finally: Is there a better way to go? Any standards, any references?