Using terminal, how can I install a DMG file without sudo rights

First of all, there are a number of commands you can try to break out of whatever weird shell state you're in. Ctrl+C, Ctrl+Z, and Ctrl+\ may exit the sudo command, and Ctrl+D would exit the 'tee.' Assuming the command is not running as your controlling shell (which it probably isn't, judging by how not entering your password gives you a shell), one of those will likely get you back to your actual shell (e.g. bash). Then you should look at the following files to see what's putting you in that state:

/etc/profile
.bashrc
.login
.bash_profile
.profile

You can also edit those files from any OSX text editor (rather than having to get into a shell to do it), although it's a bit tricky - from the "file open" dialog, press Cmd+Shift+G and then type the full path to it (i.e. /etc/profile or /Users/username/.bashrc).

It's also possible that your login shell itself may have been changed in some way. In that case, you need to go to the Users & Groups system preferences, then ctrl-click on your username, then select "Advanced Options." Ensure that the "login shell" dropdown reads /bin/bash.

Run sudo visudo (or EDITOR=nano sudo visudo) and add a line like this:

%admin ALL=NOPASSWD: /usr/bin/myscript

Then create a .command file that runs /usr/bin/myscript:

echo sudo myscript>~/Desktop/test.command
chmod +x ~/Desktop/test.command
echo $'#!/bin/sh\nls /var/root'|sudo tee /usr/bin/myscript;sudo chmod +x /usr/bin/myscript

Or if for example you want to run sudo purge without having to enter a password, add

 %admin ALL=NOPASSWD: /usr/sbin/purge

to sudoers.