How can I install software in Linux with no yum and no wget?

Do you have links or curl? Do you have ssh? Then you have scp, you can transfer for example a tar.gz source file to it and see if you have make and gcc, and build/compile the package on the machine, then set the prefix to somewhere under your home folder and install the package there. You could download and install pacman - then you have a packet manager!

To check the version of the distro usually its in /etc/release or /etc/version or /etc/release-version

But it seems to be an old RedHat machine, so you probably only have rpm. So then use scp to transfer a .rpm package to it (which sutis the redhat version) and install it with rpm -U something. Probably look into scp'ing in a wget .rpm package, after that the rpm should work better.