I have two GV accounts, one set up in the US and one set up somewhere else (I don't remember where). With the first, I'm not charged for calls to US, from France and Thailand IP addresses. With the second, I'm hit for $0.01/min.
You will find OBi devices useful for other pieces of your communications puzzle. If any of your homes have landlines or pseudo-landlines (ISP or other locked VoIP), consider the OBi110; it's only a few dollars more.
Receiving US calls on your SG mobile: GV won't forward directly to an international number. The simple option (not involving an OBi) is to get a US DID from a VoIP provider and set it to forward to your SG mobile. Possible choices are Anveo ($0.99 or $1.99/mo. + $0.018/min.) or Localphone ($0.99/mo. + $0.01/min.; IMO lower quality, some CLI delivery issues). Alternatively, have an OBi receive the call and bridge it with your choice of provider. If you have a service such as mio Voice, the calls would be free. Otherwise, depending on desired quality and simplicity, good options might be GV, Anveo, Voxbeam or Betamax, ranging in price from $0.02 down to $0.003/min. Betamax offers some "free" calls, but IMO the requirements are not a good match for your application.
Making US calls from your SG mobile: You can set up the OBi so that you call it and hang up. Based on your caller ID, the AA does not answer calls you back (with a service as described above); you can then make an outbound call on GV.
Receiving SG calls in US: If you have a landline or SG VoIP service, the SG OBi can bridge to your US OBi at no cost, or via GV to your US mobile (free but uses cellular minutes). You can get SG DIDs from many sources; an inexpensive choice might be CallWithUs ($3 setup + $4/mo.) If you need incoming from only a few contacts (recognized by their caller ID), you can use Rebtel together with any service offering free iNum, e.g. Callcentric. Any of these methods have no per-call cost to you, but the caller may be charged airtime or local call cost.
Making SG calls from US: From your OBi phone, this is straightforward. From your mobile (if you want lower cost than GV's $0.02/min.) you can call into the OBi and use the AA to bridge to another service.
If your US cell phone plan is not unlimited, but includes unlimited calling to/from specific numbers (F&F, MyFaves, A-List, etc.), bridging international calls through the OBi will avoid using minutes on those calls.