Hmmm. Well, Puppy doesn't really install as such. Not in the same way that a standard distro installs at least.
As to your download issue, are you selecting the right path for your download?
I think you would have to increase the size of the partition you originally used as a backstore to increase your personal storage space. But you should be able to tell the browser to put your downloads wherever you want them.
You tried creating a new partition, but Puppy is ignoring that and treating it as an extra drive instead. (which is normal)
Puppy uses a virtual ram disk as it's root directory, and it's using the home partition on that disk rather than an actual disk. If you have no swap partition, then there is nowhere to put the data once the ram disk is full. You could delete that empty partition and create a new swap partition in it's place to see if that helps.