Letter’s found at new construction sites

As a former construction estimator, I would suggest that you get in touch with the contractor sales department at a building supplies store in the area in which you're considering building. Quite often the estimators there will have a ballpark "per-square-foot" price for materials, and may have an idea of a ballpark price for labour, too. They can probably refer you to reputable builders as well.

That said, there are really far too many variables for a ballpark estimate like that to be at all useful. You can get a house to lock-up fairly cheaply, but it's the finishing work that's really going to determine the end cost. The cost of a beautiful metal tile roof is going to be a lot more than 20-year asphalt shingles, and the cost of vinyl siding is going to be a lot less than stone, as examples.

To get any kind of accuracy in an estimate, you're going to need to have a set of plans and you'll need to decide what exactly you want in terms of finishing, land, appliances, etc. Anything shy of that is really just going to be pulling numbers out of thin air.

Once you have corrected your axis of rotation it should be obvious that the idea to lengthen the "clip" will not change anything with regard to the force required to raise the flap. Actually it could make things worse if the longer clip added more weight to the whole assembly.

The force needed to raise the flap is measured in some units like foot-pounds (ft-lb). In your case the weight of the flap and connecting mechanism is distributed over a distance so the actual formulas to calculate this become in integral in order to solve. You could make a huge simplification and assume that your weight is all at the maximum lever arm distance and this have things over designed but it makes the analysis much simpler. So guess that you have the 5 pounds of flap + maybe 1 pound for the linkages so 6 pounds total. Assume a worst case lever arm of two feet. This means that for simple analysis that you would need a torque capability in 2' x 6lb = 12 ft-lb. That torque can be directly translated to the units that are used by the motor / gearbox combination.

Some general comments about the design.

1) If the overall design as shown is feasible I think you would want at least two if not three mount points to attach the flap to the shaft.

2) You would want to do everything possible to get the hinge edge of the flap as close to the shaft as possible.

3) You should start planning right now for some type of limit mechanism to control the up and down range of travel of the flap. That could range from direct limit switches to physical stops and detection of increased load on the motor when the stops are hit.

4) This type design calls for a gear box of some type to increase the mechanical advantage given to the motor. A worm gear drive can multiply torque a whole lot in a single stage of gearing. A spur gear arrangement may take several gearing steps to keep things in a realistic size volume.

5) Another reason you need a gear box is to reduce the speed of the motor down to a realistic motion speed for the flap. At 2-3 seconds for the flap operating through 90 degrees of operation you can see that for a motor that may want to operate at say 1750 RPM that you are going to need some gearing down. 1750 RPM for some AC motor is ~30 revs per second. Flap is 1/4 revolution in 3 seconds so corresponds to 1 rev in 12 seconds (1/12 rev per second). The gear ratio to reduce this speed is 30 * 12 = 360 to 1.

Now knowing your torque requirement at 12 ft-lb you could estimate that you need a motor of about 1/360 of that torque or about 0.033 ft-lb at the motor. Figure that gear boxes have some torque losses just to operate then so maybe a motor of 0.05 -> 0.07 ft-lb.