is it possible to connect regular objects and custom objects from Enterprise edition of Salesforce Sales Cloud to Group edition of Salesforce Sales Cloud through Salesforce to Salesforce functionality?
Salesforce to Salesforce requires API access. Normally, it doesn't have API access on it's own. However, it's possible for Group Edition to obtain API access through ISV Apps from the App Exchange. However, Group Edition does not have Sharing. That would seem to rule out the possibility of implementing Salesforce to Salesforce in Group Edition. I don't have a Group Edition Org available to me to see if it's there, but off-hand, I don't see a way of setting it up via apex if it's not already part of the built-in features.
According to the following feature comparison chart (http://www2.sfdcstatic.com/assets/pdf/datasheets/DS_SalesCloud_EdCompare.pdf) the Salesforce to Salesforce feature is available in Group Edition so it might be worth setting up a test to see if you can do what you are trying to accomplish.
I tried and tested setting up a connection between Group and Enterprise and it throws : INSUFFICIENT_ACCESS_ON_CROSS_REFERENCE_ENTITY, insufficient access rights on cross-reference id:  error while inserting PartnerNetworkCOnnection Record. However, you can share a record manually. But sharing through apex is not possible, I guess.