Thank you for the reply!

I thought about placing the old town next to the port, but I ran into the problem of the low class districts needing to be placed bordering it.
The placing of the left and top-right roads leading to their respective exit is somewhat essential, because the roads need to run through the town, which would not work from the north due to the ocean being south.
With that road placement, the lower right corner of the city would seem the "safest", meaning it would take the longest for invading armies to reach it from the nearest gate. It would feel strange to place the old town there, because as you mentioned, the more defensible parts of the city tend to be more valuable, so a low class district there would have been quickly converted to a higher class district.

The way I pictured the town growing was the old town being more of a farmland trading community in the very beginning. The dragon choose to inhabit the land because of three major factors:
It is good for trade, lying along the major road leading through the country. It is also connected via the docks.
It is defensible, with a mountain providing tactical advantage and one back to the vast ocean, meaning an invading army would need to bring ships, lest they wouldn't be able to completely cut off the citys supplies during a siege.
It has a central geographical position in the kingdom, meaning it would be easy to dispatch troops to numerous far off places of the lands not easily accessible should you have to walk cross-country.
The dragon nested in the mountain, and in the following 2000 years or so the kingdom grew under his rule.

Actually, your questions really got me thinking. Most likely, the entire low class district was "Old town" (or just "town" from the view of the first inhabitants). And it wouldn't make sense for the old town to be situated so far from the roads.
What I think I'll go with is that the low class district was the first settlement there. As the city grew, it was not so much that the low class district was falling apart but rather everything next to it was built to a much higher standard. The low class district is merely the name for the part of old town that was actually inhabited. Subsequent attempts to improve the low class district met unexpected resistance, not least from the thieves guild.
They thrive under dirty, lawless, every-man-for-himself conditions, and had much to profit from being able to keep the low class district low class, free of guards and other rulers.
Rather than driving out and purging basically everyone in the low class district (with little to no chance of actually catching the thieves guild leaders), a wall was put up. That way, most people in the city live comfortably believing that the low class district is contained from the rest of the city. (Of course, the inhabitants there are not prisoners, they are free to walk the city. They just need to pass a few guard booths to do so.)