How do I stop Google Play Services from being automatically installed? I've an old rooted Android 2.3 phone which comes with "Android Market" that is obviously updated with latest "Google Play Services". There is no "Google Play Services" installed on phone and strictly speaking I don't need it. But recently I've found that "Google Play Services" is automatically installed when connected to internet!

The problem is that phone becomes very slow (no issue with free space but I don't know why it becomes very slow after having play services). So, I've to uninstall "Google Play Services" but it is not the solution as it is automatically installed without prompting (say secretly).

As far as I know it is the "Google Play Store" which can do this as it has power (aka back door).

So, How do I stop or prevent "Google   Play Services" from being automatically installed? Or I've to uninstall "Google Play Store"? Or you can suggest the work around for the issue of phone's being slow down with "Google Play Service".
You can check this thread over at XDA-developers. You'll want patched GSF to skip self-updating. 

Please note that sticking to older GMS versions (or not having them at all!) might prevent some "modern" applications from working correctly. 

p.s. your slowness issue is more likely about low free RAM than disk space.