Install application on external SSD and load from external SSD to RAM?
Yes. Some applications won't allow you to install them on drives that don't contain an OS X installation, however there are often ways to do it manually post-installation if that is the configuration you wish.
Some applications will indeed do some writing to the internal SSD, since presumably that's where OS X is installed. The Library folders for example contain configuration settings and application data for most of the applications on your computer. Although you can trick an application into writing to your external SSD by moving its data/configuration directory onto your external drive, then making a symbolic link back to your Library folder. It's not guaranteed to work, but it generally works just fine.
An application whose data is entirely contained on the external SSD won't have any reason to access the internal one. The data would load directly into RAM, never once touching the internal SSD.