The check for updates lags behind releases, often by a few weeks or more, to give a “beta-test period” before indicating to the general public that an auto-update is available; the developer doesn’t set it to auto-update to a new version until he’s more comfortable that it’s reliable.

You can always download and install the most recent version at https://notepad-plus-plus.org/download/ , if you are willing to participate in the “beta test”.

update: there were some known issues with 7.6.0 and 7.6.1, which is why there was such a quick succession to 7.6.2. If you want to use the new folder structure @Meta-Chuh mentioned, you have to be on 7.6.2.
Also, your Debug Info (thanks for that) does confirm you are in installed mode, not portable (Local Conf mode: OFF indicates installed; ON would indicate ‘portable’)

Debug Info:
Notepad++ v7.6 (32-bit)
Build time : Nov 12 2018 - 23:51:42
Path : C:\Program Files (x86)\Notepad++\notepad++.exe
Admin mode : OFF
Local Conf mode : OFF
OS : Windows 7 (64-bit)
Plugins : none

I do not believe I am using the portable version. I am only using 7.6 and when I click on “update Notepad++” I am advised there is no update. When I check the %ProgramData% folder, there is no Notepad++ folder there - so I am guessing it would update if I moved to version 7.6.2 - but not sure how to do this when the Update Notepad++ says there are no updates available.

Also, currently the program is installed on Windows 7 in Program Files (x86). I will be getting a new Windows 10 laptop in a couple of weeks, so maybe I should just install the program fresh on that laptop and not worry about this one.

if you have recently upgraded to notepad++ 7.6.2 (recommended), the location and folder structure, where the plugins are supposed to be, have changed, and you need to move them to the new location.

installed notepad++ version 7.6.2:
the new location for an installed 7.6.2 version of notepad++ is :%ProgramData%\Notepad++\plugins which will resolve to C:\ProgramData\Notepad++\plugins on most systems.

taking ComparePlugin as an example, the folder structure should be:
C:\ProgramData\Notepad++\plugins\ComparePlugin\
  containing:
  ComparePlugin.dll
  ComparePlugin (folder)

C:\ProgramData\Notepad++\plugins\ComparePlugin\ComparePlugin
  contains:
  git2.dll
  sqlite3.dll

note: the folder structure for portable versions is similar, but confined within the portable version’s program folder.

portable notepad++ version 7.6.2:
again taking ComparePlugin as an example, the portable folder structure should be:
npp.7.6.2.bin\plugins\ComparePlugin\
  containing:
  ComparePlugin.dll
  ComparePlugin (folder)

npp.7.6.2.bin\plugins\ComparePlugin\ComparePlugin
  contains:
  git2.dll
  sqlite3.dll

one strange thing you mentioned, is that you don’t even see the plugins menu.
are you using a portable version or an installed version ?
which notepad++ version are you currently using ?
please also go to the notepad++ help menu: ? > debug info and paste it here