There's not really a need for anything dramatic beyond directly drawing onto Cartesian coordinates. Simple heuristics can be used to handle the pathing and likely to hit the optimal minimum number of angles the majority of the time, but and likely the shortest length path even more often. All of this can be done dynamically as you require, but while maintaining precision of graphics without breaking the screen up more discretely that it needs to (pixels should remain the most discrete level) and without the need for complex algorithms.
For the overlay, just set all pixels to the color of your lines and modify the alpha channel bits to transparent or opaque depending on if the pixel is or isn't part of the line. To figure out which bits that are part of the line requires a bit of geometry, but that's a piece of cake once you have everything in place.
To figure out how to draw your line onto the alpha channel, you'll need to figure out the style of your lines. A lot of what you'll do depends on style. A common style is using straight lines that are horizontally and veritcally aligned with quarter circles for right angles.
For the "avoidance" algorithms, these aren't too difficult to implement when you just want to avoid the "boxes" representing your nodes... to decluster all your lines is a bit larger of a task and something that not even Visio employs. To avoid the boxes/nodes, using the midpoint between the edges of the box (such as the vertical edges between geo1 and geo3) is nice to do for symmetry and then picking a simple predefined distance to keep non-connecting lines (that is lines that do not connect to that particular box) away from the boxes also works well. A generalized algorithm for this is simple to do, but a little too verbose to describe here, but is essentially a set of generalized checks and switches working on horizontally and vertically align lines and quarter turns. If you end up wanting more detail on how to do this, just post a comment on this answer.
If you're looking for something that's already made for you, the type of connections and rearranging that you want really depends on the application and not a lot of people make tools that are low in demand or too specific of a demand. Obviously this type of software is out there since Visio and others employ it, but whether or not it's available as open source or part of some other free libraries I'm not certain.
