Answer:

FOXHOUND3 is very important. It is set by the Foxhound InstallShield setup program to contain the drive and path of the folder where Foxhound is installed. It is then used in most of the Windows command files that are executed by the Foxhound shortcuts. For example, the "Start - Foxhound3 - Start Foxhound via Chrome" executes the $start_foxhound3_chrome.bat file which contains the command CD /D "%FOXHOUND3%".

Normally you do not have to enter or change the FOXHOUND3 environment variable, but if you do, here's how:

Start 
  - Control Panel 
    - System 
      - Advanced system settings 
        - Environment Variables... 

FOXHOUND3BIN is optional and rarely used. You can set it to Bin32 if you want to start Foxhound using the 32-bit version of SQL Anywhere 16 even if the 64-bit version is available. Values other than Bin32 have no effect. The default action (if FOXHOUND3BIN is not set to Bin32) is to start Foxhound using the 64-bit version of SQL Anywhere if it is installed, otherwise use the 32-bit version.

No automatic process sets or changes the FOXHOUND3BIN environment variable, it's all up to you:

Start 
  - Control Panel 
    - System 
      - Advanced system settings 
        - Environment Variables... 

FOXHOUND3UPGRADE is optional and rarely used. If it is set, it is used as the initial value for the FOXHOUND3UPGRADE setting in the post-setup process the next time you reinstall or upgrade Foxhound.

The post-setup process will prompt you to provide a different value if one is desired:

Post-Setup Process for Foxhound Version 3.0
*** Checking for a post-setup path parameter...
*** A post-setup path parameter was provided...
C:\ProgramData\RisingRoad\Foxhound3\
*******************************************************************
***                   Foxhound 3.0.4380
***
*** Here's where Foxhound is being installed:
*** C:\ProgramData\RisingRoad\Foxhound3\
***
*** Starting the Foxhound 3.0.4380 post-setup process...
*** Creating foxhound3.db.3.0.4380.ORIGINAL_COPY...
*** Checking for an existing Foxhound3 installation...
***  ...yes, there is an existing Foxhound3 installation.
*** Checking if the existing data should be upgraded...
***  ...yes, the existing Foxhound3 data should be upgraded.
******************************************************************
*** PLEASE READ THIS, AND CONFIRM OR CHANGE **********************
******************************************************************
***
*** "FOXHOUND3UPGRADE" specifies how much data is to be upgraded.
***
*** If you want to CHANGE the setting, type in a new value...
***    ALL       - upgrade all the data
***    OPTIONS   - no samples, just the Foxhound options
***    yyyymmdd  - options plus samples since yyyymmdd
***    nnn       - options plus last nnn days of samples
***    NOTHING   - don't upgrade any data
*** and press Enter to continue.
***
*** If you LIKE the current setting...
***    FOXHOUND3UPGRADE=ALL which means upgrade all the data
*** just press Enter.
******************************************************************
Current FOXHOUND3UPGRADE=ALL
New     FOXHOUND3UPGRADE=

The default action (if FOXHOUND3UPGRADE is not set to any value) is to use the the value ALL as the initial value, which you can change as shown above; see How do the different FOXHOUND3UPGRADE values work?

No automatic process sets or changes the FOXHOUND3UPGRADE environment variable, it's all up to you:

Start 
  - Control Panel 
    - System 
      - Advanced system settings 
        - Environment Variables... 

See also...
Why isn't the new value of the FOXHOUND3 environment variable available immediately after Foxhound is installed?
How do I tell if Foxhound is running on the 32-bit or 64-bit version of SQL Anywhere?
How do the different FOXHOUND3UPGRADE values work?