Jeithorpe

The use of Power duplicate (or Move data entry) and Compounds, with pertinent snapping options, could help you to "automate" the creation of regular shapes. 

I don't think it's possible. – But the Cog Tool seems to be suitable for that shape:

I assume to achieve "1000% perfect precision" for a shape, we would possibly have to calculate all angles and lengths mathematically correctly and then construct the shape as a composition of the simplest possible shapes (geometric addition, subtraction, etc.).
Just in case: There are several threads in the forum about mathematical precision versus the generic Affinity Shape Tool objects. For example, an Affinity circle is not a "perfect" circle, FWIW.

The problem is here:

I would redraw it, because correcting this seems to me more difficult than doing it right at once. Use of snapping and mirroring takes some trial and error, but finally proves to be helpful. 
 

I'm sure there are several useful ways to construct such an object. For example, your starting point with the "Square Star" shape may work, too. And rotating the object might also work or at least wouldn't harm if the object has a 1:1 aspect ratio with the individual edge segments of the polygon having identical length.

What limitation do you mean? And what rotation at all? – I guess I'd start with a hexagon. Possibly it helps to define a desired/ the required grid first. It also may help to use the button "Cycle Selection box" (base/regular bounds) in the Context Toolbar. And zoom in for more precision.

As workaround – e.g. if the previous calculation of lengths or angles isn't 100% perfect – a last option might be to convert to curves + move nodes to match the required edges. But once you start moving individual nodes, it can easily make things worse.

@Jeithorpe, welcome to the forums
My first guess would be something is preventing access to the internet? Check your network settings to see if the computer is able to access the internet. Can you open a browser and access the internet?
It could be a firewall issue, blocking Affinty.