That makes no sense to me at all.

My understanding is android syncs your google contacts. Thunderbird has a Google contact sync add-on. Is that what your talking about? https://addons.mozilla.org/en-US/thunderbird/addon/gcontactsync/

Installed Thunderbird AB Sync to work on Android cell phone

https://play.google.com/store/apps/details?id=com.rogerang.android.TBABsyncFree&hl=en

in order to sync my emails on my PC with emails on my cell phone. In other words to have Thunderbird on my cell phone be in sync with the emails on my PC.

In order for it to sync the emails on my cellphone with those that I receive on my desktop PC running Thunderbird under Windows 7, it seems that it is necessary to get a key that is obtained from Thunderbird running on the PC. Supposedly the way to get the key is to go to Thunderbird extension in Thunderbird on the PC and select: Tools>Add-Ons>Extensions>Android Contacts Sync>Options. I am able to go to tools, add-ons and extensions, but do not see "Android Contacts Sync" Only ones I see: Extension List Dumper 1.15.2.1 Extension Options Menu 2.17 Lightning 4.0.4.1 Theme Font & Size Changer 42.5 McAfee Anti-Spam Thunderbird Extension 2.0

Should I give up on Thunderbird AB Sync or is there some other tool that would allow me to use Thunderbird on the cellphone?

You do not get email from Thunderbird. You get it from an email provider. All cell phones come with an email client installed. Add the details for your actual email account to that client and be sure to use IMAP.

IMAP email is designed to use when checking email from multiple devices. The email account must be capable of IMAP and all your clients setup as IMAP.

Should I give up on Thunderbird AB Sync or is there some other tool that would allow me to use Thunderbird on the cellphone?

Yes. Thunderbird does not run on phones or tablets and you seem to be going about syncing email the hardest way possible.

FYI: The add on your are messing with is for syncing your address book, not email messages.