2.2 froyo - Can't install Google Play Services

I just upgraded Google+ to 3.6.0.44338848 (released late March 2012) and now it won't launch. It says:

This app won't run without Google Play services, which are missing from your phone.

If I tap "Get Google Play services" I'm brought to Google Play where I tap "install" and "accept and download" but then I get "Error: No connection."

(I do have a connection. I can browse the web just fine.)

Now, I should mention that Market/Play has been odd the whole time, I've had this phone, a T-Mobile myTouch 4G running Android 2.2.1. Whenever I want to install an app, I have to go into Applications, tap Google Play Store, and tap "Uninstall updates" to restore the Market app. Then I can install applications. Over time, some process upgrades Market to Google Play but I can't install applications until I downgrade to Market to avoid the "Error: no connection" message. The processes is frustrating. On my old LG phone (also running 2.2 Google Play worked just fine).

I'm been putting up with all this downgrading because it's not often I install apps and usually there's no problem after installing or upgrading an app.

But now the Google+ app is unusable so I'm motivated to do something about the "No connection" problem.

I've read through posts such as http://www.teamandroid.com/2013/01/14/fix-google-play-store-no-connection-retry-error-message/ and Google's help page at How to access the Google Play Store app - Google Play Help but I'm stuck. Please help.

Answer

I was able to install Google Play Services by browsing to its page at https://play.google.com/store/apps/details?id=com.google.android.gms and then logging in, clicking install, and choosing my phone.

Come to think of it, I might start installing more apps this way, from the comfort of a desktop browser. :)

Anyway, I'm able to use the Google+ app again.

I'm still puzzled by the "No connection" business and having to periodically downgrade Play to Market to install anything from my phone running Android 2.2, but I'm glad I didn't have to root it and install CyanogenMod as suggested by some friends.