I know nothing about Connect-Me, but I do have an SPA3102 and have had no trouble getting it to disconnect. When the caller hangs up, does the Connect-Me play a "disconnect tone"? If so, on the SPA's PSTN Line page, set Disconnect Tone to match. It's possible that the box interrupts loop current (CPC) or reverses polarity, in which case enable the corresponding option. If all else fails, you can set the SPA to disconnect when there is a specified period of silence.
For caller ID, confirm that you have a sufficient PSTN Answer Delay, and that the caller ID settings on the Regional page match what the Skype box sends.
However, IMO Skype (as a way to connect to the PSTN) is a poor value for the vast majority of users. If it seems to be a good deal for you, please post some details about where and how much you call, and how much you are spending with Skype.
You might try Google Voice (free to US and Canada; free incoming; see »wiki.freeswitch.org/wiki ··· le_Voice
). I'm reasonably certain that the module doesn't depend on access to audio devices. For straightforward SIP (PAYG) service, look at Callcentric, Anveo, VoIP.ms, or Localphone, listed IMO in decreasing order of price / quality / reliability / support. For heavy "unlimited" use, check Phonepower, VOIPo.