Nizdobs,
You can of course easily accomplish this with the OBi.
First I would look for available incoming U.S. numbers in the area code/rate center that you wish. Companies I would look at that might provide the incoming number would include CallCentric,
Voip.ms, Anveo, Future-Nine, GoogleVoice, LocalPhone. Some companies give free incoming calls, some companies have per minute charges for incoming calls, some companies offer both options and vary the monthly charge.
Once you find the incoming number then some companies require you to use their outgoing service, other companies will let you forward the incoming number to a voip account at another provider without additional cost via a sip uri call. In most cases you usually use the voip provider that is providing the incoming number.
For outgoing service if caller id is important, and I imagine it would be for the type of operation you described, then you need to make sure the outgoing call will show the incoming number you have selected. The other part of caller id is the name. Some companies will include the name, some include the name for a fee, others do not include the name. It is a question to ask.
The cost of the incoming number may be included in the outgoing service or it often will be separate. You probably would also want to avoid having to pay an additional fee for 911 emergency calling since you will be using the number outside the United States.
Outgoing calling can be with a monthly plan or more often pay-as-you-go. Monthly plans usually say unlimited calls which really means not excessive or fair usage which means the same thing although for competitive reasons they usually don't publish the number. Monthly plans often specify for residential use only. For pay-as-you-go you deposit money up front and outgoing calls are charged by the minute. Many providers will offer an optional plan to automatically debit your credit card when you get low. Pay-as-you-go outgoing calls to the U.S. and Canada usually will run between USD $0.005 per minute to $0.02 per minute depending on the voip provider.
Probably the largest voip providers would be Vonage, Skype, and GoogleVoice. You wouldn't use an OBi adapter with Vonage or Skype. Vonage offers monthly plans with good quality communications but you need to use a voip ata provided by the company and many think their service is overpriced. Skype offers monthly plans and reasonably good voice communications but uses their own proprietry protocol, requires a computer and/or special phone equipment to use the service. Skype lets you setup a caller id that they confirm. I'm not sure how that would work out for you. GoogleVoice is attractive to many because it offers free U.S./Canada calling, will work with an OBi ata, but the terms and conditions say it is for residential use only and you may wind up without service at some point in time. GoogleVoice will show a caller id of your GV number without a name.
My experience with calls to/from Chile where it is calling to / from a Chilean PSTN number is that the voice quality can vary greatly depending on the time of day and the voip provider. I believe this is due to the available internet bandwidth from the U.S. or Europe to Chile and/or the company used by the provider for call termination. Your case is a little different in that you are not terminating the call thru a Chilean PSTN company, just thru the available bandwidth directly to your Chilean location (ip address). I am sure you are after the highest quality at a reasonable price and I think you may want to try different providers to settle on one that you believe to be satisfactory. If possible, I would try to setup a couple of providers, one for backup.
The location in Chile will, of course, need a good high speed internet service. If a good internet connection is not possible then there are more complicated ways to attack the problem using a local Chilean pstn number.