Answer:

FOXHOUND1 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 - Foxhound1 - Start Foxhound via Chrome" executes the $start_foxhound1_chrome.bat file which contains the command CD /D "%FOXHOUND1%".

Normally you do not have to enter or change the FOXHOUND1 environment variable, but if you do, here's how:

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

FOXHOUND1BIN 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 12 even if the 64-bit version is available. Values other than Bin32 have no effect. The default action (if FOXHOUND1BIN 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 FOXHOUND1BIN environment variable, it's all up to you:

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

FOXHOUND1UPGRADE is optional and rarely used. If it is set, it is used as the initial value for the FOXHOUND1UPGRADE 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:

******************************************************************
*** You can control how much of the existing data is upgraded.
*** To do that, you can set the FOXHOUND1UPGRADE environment
***   variable BEFORE running this upgrade process,
***   OR you can type in a temporary override here.
***
*** The current setting is...
***   FOXHOUND1UPGRADE=ALL which means upgrade all the data
***
*** 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.
*** You will be asked to "CONFIRM OR CHANGE" again.
***
*** If you LIKE the current setting, just press Enter.
******************************************************************
Current FOXHOUND1UPGRADE=ALL
New     FOXHOUND1UPGRADE=

The default action (if FOXHOUND1UPGRADE 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 FOXHOUND1UPGRADE values work?

No automatic process sets or changes the FOXHOUND1UPGRADE 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 FOXHOUND1 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 FOXHOUND1UPGRADE values work?