Connecting iOS app to Windows/Linux apps
I would definitely suggest a webapp. And the answer to your questions are given below:
How would I receive and send notifications over a local network.

Use a REST based web service to communicate with the server. 
  You have to use polling to receive data:-(

How could I connect to the server using NSURLConnection if it does not have a static ip?


If possible configure a domain name in your network which points to server ip. (Configure local DHCP to give same IP to your server every time based on mac address!)
Have a IP Range and when the app starts, try to reach a specific URL and check if it is  responding.
Ask the user to enter the server IP every time the app starts!
Background:
I am just about to start development on an mobile and desktop app. They will both be connected to a local wifi network (no internet connection) and will need to communicate with one another. At the outset we are targeting iOS and Windows as the two platforms, with the intention of adding Linux, OSX, and Android support in that order. The desktop app will largely be a database server/notification center for receiving updates from the iOS apps and sending out the data to other iOS apps. There may be a front end to the desktop app, but we could also incorporate it into the iOS app if needed.
For the moment we just want the iOS app to automatically detect when it is on the same network as the server and then display the data that is sent by that server (bonjour like).
As far as I see it there are two paths we could take to implement this

Create a completely native app for each platform (Windows, Linux, OSX). 
Pro: We like the ideas of having native apps for performance and ease of install.
Con: I know absolutely nothing about Windows or Linux development.
Create an app that is built using web technologies (probably python) and create an easy to use installer that will create a local server out of the desktop machine which the mobile apps can communicate with.
Pro: Most of the development would be cross-platform and the installer should be easy enough to port.
Con: If we do want to add a front-end to the server app it will not be platform native and would be using a css+html+javascript GUI.

Question:
My question is how would  implement the connection between the iOS app and server app in each circumstance.

How would I receive and send notifications over a local network.
How could I connect to the server using NSURLConnection if it does not have a static ip?


I hope this is clear. If not please ask and I will clarify.
Update 09/06/2013
Hopefully this will clear things up. I need to have a desktop app that will manage a database, this app will connect to iOS devices on a local wireless network that is not connected to the internet. I can do this with either the http protocol (preferably with a flask app) or by using a direct socket connection between the apps and the server. My question is which of the above two choices is best? My preference would be for a web-based app using Python+Flask, but I would have no idea how to connect the iOS app to a flask app running on a local network with out a static ip. Any advice on this would be appreciated.