Want to buy but have questions

You may be able to use the Belarc Advisor to find the serial number:

Products: Belarc Advisor

Have you tried registering the software, now, and see if that makes the serial number show up?  Registering the software ties your AdobeID to the serial number so that you can look it up on your Adobe account, but you may not need to know the serial number to register it, it may be transmitted automatically.

The main issue would be what you mean by PSE10 came pre-installed on the computer 5 years ago with the problem being if this computer belonged to someone else, an individual, and you purchased the computer from them, but the PSE 10 license belongs to them.  If so then they would have to transfer the license to you using an online form at Adobe.

Or if the software was something bundled with a new computer, it may have documentation telling you the serial number and how to register it or it may be some special version of PSE that was sold to the computer manufacturer for a cheaper price and as such cannot be used as the source of an upgrade.

Basically, the software needs to be not registered to anyone else and be a version that isn't some promotional version that cannot be upgraded or installed again, and you can discover the serial number, then you may be able to install it, again, using that serial number.

If you find the serial number ,the download links to the older version of PSE10 should be here:

Adobe software and other downloads

Yes you can with the key for PSE10, if you have only ever installed on one computer. You are permitted two activations.

Or you could get the latest version PSE 2019 at the upgrade price if you still have the original serial number. But be aware that black Friday discounts towards the end of November often mean the full version becomes cheaper than an upgrade for a limited period; especially at on line stores.

In fact you could download PSE 2019 right away and use it as a free trial. If it runs ok on your laptop there would be no need to un-install. Simply activate with the serial number from disk or keycard, after purchase.

Go to the bottom of this page to download:

https://www.adobe.com/products/elements-family.html