To update a installation database go to Puppy Package Manager, click on the little button on the upper left that looks like crossed tools, then choose the folder tab at the top that says "Update Database". In the window that opens click Update Now. Answer yes to any questions about updating. The package manager should restart when it finishes. Click the little tool icon to go back to the main installation screen (if you aren't returned there already), and see if you can now load the package you want.
Note that not all Ubuntu packages which Bionic Pup can install necessarily work well, since Bionic Puppy is not Ubuntu Bionic. But many will work okay.
For Browsers like Firefox, and my preference Seamonkey (also a Mozilla browser) if you want the latest version, it's best to just go to the Firefox site and download it from there. A second alternative is to look for a Puppy version of a portable Firefox. The least good method is using the Puppy package manager to get an Ubuntu version.
As for Tux paint -- I've never used it. Puppy has MTPaint already, which I use a lot. If you need something more heavy duty -- equivalent to Win programs like Photoshop, then the usual alternative here is Gimp.
I'm not a great fan of Gimp, and actually prefer old versions of some free Win programs like Photofiltre Version 7 which is available free froms sites like Oldversions. But you do have to be running Wine to use windows programs - and that's a "whole nother" subject. If you want to stick with Linux, MTPaint and Gimp are the usual in Puppyland.
ps the error message you show mentions Wayland -- that's not implemented in Puppy as far as I know, and really isn't necessary for a graphics program or Firefox for that matter, in Puppy. I'd avoid whatever that was you were attempting to download. If Firefox, just get Firefox from the sources I suggested above.