Retrieving data from extremely gummed up 2009 iMac, via iCloud, using 2021(?) iPad? If not, how can I?

There is a very good chance the hard drive is failing on such an old system. A failing hard drive can severely impact system performance. If the hard drive is failing, then you need to be careful since the more you use a failing drive, the more likely the failure will get worse to the point you won't be able to retrieve any documents.

First thing I would do is run the DriveDx and post the complete DriveDx text report here using the "Additional Text" icon which looks like a piece of paper. If there are only a few bad blocks, then manually copying the data will likely be Ok, otherwise regular apps & macOS won't be able to work due to all the errors produced by a failing hard drive. This app will have a 14 day trial period....do not purchase the app at this time since the status of the developer is in question as the actual download button on their website has been broken for months, although payments are still being accepted. This link is direct from BinaryFruit... the developer of DriveDx:

https://binaryfruit.com/download/drivedx/mac/1/bin/DriveDx.1.11.0.zip

FYI, you should always have frequent and regular backups of your computer and all external media (including the cloud) which contains important & unique data. With SSDs and newer Macs, there is usually no way to recover data if there is any sort of problem with the newer Macs and there are a lot more ways to permanently lose access to your data if it is not backed up.

Copy them to an external drive, or even USB pen, but you really should have an external drive for backups.