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Android apps don't pull in dependencies like that. But even though it's a 13 MB download, I see when installed it's showing me it takes up 76 MB, before I've even opened it for the first time.
No, I don't know why there is a discrepancy, nor why it told you 500 MB. But I do know the fancy new thing for apps is a split package, and usually apps require dynamic recompilation during installation, so that might be why they ask a safety margin of up to 500 MB. (But I agree it still seems high.)
Download size vs. Install size
It's a 13Mb download that get extracted - depending on contained data and compression algorithm you can get wild space efficiency out of some routines - and the 500Mb total make sense for an image sharing app that needs a lot of cache. The potential update mechanism also comes into play - especially update mechanisms that work with delta updates need a large amount of disk space readily available, because they construct the updated files by taking the old one plus the update information as inputs, creating the updated version of the file while still needing the old version available.
on a sidenote, having nearly no space left causes NAND based storage to degrade faster - aiming for at least 10% space left available is the number i heard and try to keep to prevent issues. This also might have been the reason your phone prevented installation of additional apps. If your storage is already near its limit, something like writing down a log file might fill it up completely, leading to instability and crashes (had this happen on my linux machine because i had downloaded a bunch of stuff and didn't watch space remaining; it's easily fixed, but also something that many casual users aren't aware of)
From my experience filling my phone's storage a bit too frequently, there's a threshold where past that, Android considers as being low on storage, and starts doing contingency procedures to not fully fill out (if it does, data may start corrupting). Don't know how short on storage you were, but from I observed on my phone at least, below ~1 GB, cached files start getting deleted, and below 500 MB, Android gives the persistent(ish) storage running out error. If you were on either of these thresholds, this may have been the reason.
13MB is the download size in GP. This is different from fDroid, because you download only a device or architecture specific installer, which is different on fdroid. After installing this extracts to maybe 75mb, which should be identical to fdroid. After first start the app, this can download additional packages, if needed and the wasted space grows. It grows more with the cache and other temporary content. A app can also reserve some space. For some apps, this could be up to 5gb and more.