I recently acquired new virtual hardware from chicagovps.net, specifically a KVM LINUX VPS with 1GB RAM, 1 x vCPU, and 30GB HDD space. After rebuilding the operating system to Ubuntu 12.04 X86 64 Desktop Gen2 V1, I upgraded it to Ubuntu 14.04.6 using "do-release-upgrade" command.
Then, I installed Ubuntu desktop and upgraded it again using the same command. Currently, I have a desktop, GUI, mouse, but I'm not certain about the Ubuntu version. Unfortunately, I'm unable to login with my credentials and am wondering how to log in or obtain browsers like Firefox and Chrome.

It is highly probable that this situation will result in an unfavorable outcome. Even with KVM, upgrading a single Ubuntu LTS version on a VPS can cause critical errors that one might not be able to fix independently. In this case, it is essential to upgrade 5 LTS versions.

My personal experience suggests that it is best to avoid such scenarios if the hosting company does not offer the latest Ubuntu LTS version two years after its release. This is the case with Ubuntu 20.04 LTS currently. The underlying reason for such challenges is the customized configuration that VPS hosting companies implement while creating the Operating System Image that they install for clients.

Although Ubuntu 20.04 LTS is recommended, it appears that your provider does not offer it officially. However, if the most recent version you can obtain from them is Ubuntu 16.04.7 LTS, it's still a better option compared to Ubuntu 12.04. Nonetheless, it's advisable to upgrade to 20.04 LTS manually as soon as possible to avoid any security risks associated with outdated software.

To avoid running startx on the VPS - which may be risky over the network - you can leverage SSH's X11 forwarding capability. Instead, install the default VNC server by using the "apt-get install package_name" command (where "package_name" is replaced by the actual package name(s)).

Finally, it's necessary to upgrade the RAM on the VPS adequately to facilitate the smooth operation of Firefox. Suggestively, 4GB RAM is an ideal starting point - going lower is possible but expect subpar performance.

These are the recommended operations you can undertake:

1. Remove Orca, which is a pre-installed application for reading screen text that may not be useful to everyone. Use the following command: "sudo apt remove orca".

2. Install a classic desktop environment that does not require 3D acceleration support. GNOME desktop environment uses 3D effects, which can affect the performance of the entire system. To resolve this issue, install the gnome-session-flashback software package by running the following command: "sudo apt install gnome-session-flashback". Once installed, log out of the system, select "GNOME Flashback (Metacity)" as the session type, and log in again.

3. Optimize the Firefox web browser's performance by altering its configuration parameters.

4. Install an additional web browser for backup purposes in case Firefox encounters issues, especially when using numerous add-ons.

To log in to your Ubuntu desktop, you can use the credentials you set up during the installation process. If you are unable to login with those credentials, it might be worth double-checking that you entered them correctly.

As for installing web browsers like Firefox and Chrome, you can follow these steps:

1. Open a terminal by pressing Ctrl+Alt+T.
2. Update the package lists by running the following command:
  ```
  sudo apt update
  ```
3. Install Firefox by running:
  ```
  sudo apt install firefox
  ```
4. Confirm the installation and wait for it to complete.
5. If you want to install Google Chrome instead, you can download the .deb file from the official Chrome website (https://www.google.com/chrome/) using the wget command, like this:
  ```
  wget https://dl.google.com/linux/direct/google-chrome-stable_current_amd64.deb
  ```
6. After the download is finished, install Chrome by running:
  ```
  sudo dpkg -i google-chrome-stable_current_amd64.deb
  ```
  Note: If there are any dependency errors, you can fix them by running `sudo apt --fix-broken install` and then re-running the previous command.
7. Confirm the installation and wait for it to complete.

Once the installation is done, you should be able to find Firefox and/or Chrome in your applications menu or launch them from the terminal using their respective commands: `firefox` for Firefox and `google-chrome` for Chrome.