Acceptable joist depth for a second set of ceiling-only joists

As @TheEvilGreebo mentions, 2x6 is not enough for a floor that will be used as "habitable space". I know because when I remodeled my attic to convert it to a bedroom, the existing 2x6 floor joists (over a very similar span) were not enough and I had to sister them all.

At a minimum you will need to check with your local (municipal or county) building department. They should be able to tell you what you need.

It might be easier to get a structural engineer or architect to come out and tell you what to do. If they are not drawing up plans you might be surprised that their prices are quite reasonable (they should be able to give you a quote for what it would cost for them to come out and generate a report up front). It will also help when you get your building permit to be able to submit the report written by a certified professional to the building department.

OK first let me say that if you're going to be making major structural changes to a building (and this counts as a pretty serious structural change in my mind) it would be well worth your while to get an engineer or other qualified building professional to help you with your design. If you just "wing it" you may be endangering the house and its occupants. There may also be legal or insurance-related ramifications to doing this outside the proper channels. However, let me offer these pieces of advice:

• Do not confuse "service loads" (the day-to-day loads on a structure) with "design loads" (the loads the structure is built to withstand — typically much higher). Whatever condition you observe on a given day is probably not the worst-case scenario.
• Deflection (stiffness) is not the only consideration when designing a structure, you must also consider its strength - i.e. how much weight it can take before it breaks. Strength and stiffness are both important.
• I don't know where you came up with the 4500 lbs quantity, it sounds somewhat arbitrary since you haven't provided any background info. The amount of weight you design the structure for is based on your local building code, which includes factors like snow loads, wind loads, etc. which may be difficult for you to estimate. The weight of the roof itself is only a portion of the load you must take into account (usually less than half). These loads are typically specified as pounds per square foot (probably in the range of 30psf to 60psf) in addition to the weight of the roof.
• Your existing 2x6 situation sounds weird, I wonder if someone added an addition and the columns didn't quite line up. I would not assume that it's OK as it is now unless you see some reliable documentation. You should design your new solution for the geometry of the house and the requirements of your building code, not just matching the existing solution.
• Stresses on a cantileved beam are more complicated than a simple span between two supports. You haven't described your proposed cantilever at all but the backspan is an important part of the design. And again, you must consider the strength of the beam, not just its stiffness.