1. What are some likely shapes that an Arcology would be built in.
I would assume that towers of some kind are the most effective structures if using the surrounding land is something the builders/designers don't want. Low-gravity worlds can afford higher structures. A society with a strong affinity for quadrilaterals might build a giant cube or set of cubes. You could draw inspiration from real Earth skyscrapers. Remember that skyscrapers/tall arcologies must have foundations a deep as a significant ratio of the height. (This should give plenty of space to explore for question 3.)
2. What would be the internal layout and arrangement of the communities. The image that keeps coming to my mind are residential "blocks" connected to one another and central hubs. Like the spokes around a wheel.
This would be dependent on the exterior structure of the arcology though with a sufficiently large interior, you can organize the interior however you like. Reading up on city layouts in the context of Cities:Skylines may give you some ideas about how to adapt the structure to the needs of your story.
3. What would the "bowels of an Arcology be like. would it to badly strain plausibility for them to be "Absurdly Spacious". Dungeon crawls are essential parts of many stories and games.
Absurdly spacious is a good description for it. Infrastructure to support all those people would be distributed throughout the structure with large tunnels to connect them all. For example, if the powerplant for a community block goes goes offline for maintenance, then you'll want to get power from other plants. Designers would put in the same kind of redundancy for sewage, water, air, cargo transport, and human transport.
As a way to generate the dungeons you wanted, the arcology would have been built with large empty spaces for future adaptation and for structure reasons (hollow tubes are stronger than rods). Over time, new powerplants, sewage treatment, living spaces, etc would be constructed in the empty spaces and the older infrastructure taken offline and abandoned because of the cost of extraction/repurposing/lack of time. "Things" could move into those abandoned spaces and tunnels.
4. How cost effective are arcologies vs cities. one of the possible usages that I have for them is as a tool of planetary colonization. just land a prebuilt arcology on a world and a lot of the work of setting up a colony is done.
I think arcologies are crazy effective compared to cities. Note in this map that the densest population centers are also the highest economic powerhouses? New York is 469 mi^2, 8.4 million people and an economic output between 1.0 and 1.4 trillion dollars. Economies of scale kick in the denser you can pack people in. A well managed arcology with a population of double New York City could easily create as much economic output, perhaps much more. (Not an economist so I can't give you any numbers to go by.)
Your civilization is going to be pretty advanced to build something that large in the first place and crazy crazy advanced to land a gigaton or teraton sized structure in the gravity well of a planet. You have to respect the surface area to volume ratio.
5. How are property rights handled with in an arcology? Space being leased from the government or corporation that owns the arcology is what I'm seeing. Something akin to stores renting space from a mall.
Your suggestion about how to divvy up space in the arcology seems reasonable to me. The rent/ownership system in Europe and the US is several centuries old and has held up well. I think that's a reasonable model to work with.
6. How does the economy work in an arcology?
Currently, my favorite book on income inequality and the distribution of capital is Capitalism in the 21st Century.
Additional Reading on super-large buildings to house billions of people. How this author goes about "designing" a building to house all 7.3 billion people on earth might serve as a good model for when you design your arcologies.