First the simple answer–Yes, on modern engines thrust must be at idle and the aircraft on the ground to unlock the reversers. After they are deployed the engines can be spooled to supply “thrust”
Next for the amount of reverse:
You are correct in that in modern bypass engines only the bypass air is reversed. This accounts for around 75 to 80% of thrust put out by the motor. This of course varies slightly between designs, but its a good ballpark. Some older straight turbojet and low-bypass engines did have clamshell reversers that did redirect 100% of the air (see the MD-80 series for a good example).
You are also correct that the air is not redirected at a perfect 180 degree angle, so it is not a true “reverse” but more of a forward redirect. As far as the amount and effect of that, it is beyond my knowledge and a question for the engineers.
Some other notes on reverse thrust–
It is surprisingly ineffective at slowing the aircraft. It certainly helps, but the biggest benefit is from cancelling the forward thrust more than actually reversing. In fact, the Cessna Citation CJ series has a set of “thrust attenuators” (small flaps behind the engines that block forward thrust) in place of reversers…they are surprisingly effective.
Some airlines have the policy to only pop the buckets at idle reverse and not accelerate the engines after landing (unless needed) to save on wear and tear. The theory is that it is cheaper to replace brakes than to incur extra cost of increased reverser overhaul, engine wear and fuel expense for little benefit. Others want full reverse and minimal braking until needed. Just depends on what parts are leased vs owned and what the accountants say is cheaper. Pilots of course always have the option to use full reverse if needed.
Reverse thrust is aerodynamically most effective at high speeds. This is kind of works against you. In a short runway situation, by the time you touch down, wait a second or two for the aircraft to go into ground mode and allow reverser deployment, select reverse, the reversers to unlock and deploy and finally a few seconds for the engines to spool back up, the spoilers and heavy braking have slowed the aircraft significantly thus negating much of the benefit. Of course, every bit helps, especially on a short or contaminated runway.
Reverse thrust is generally not included in landing and rejected takeoff distance calculations. Any benefit is a bonus.
Hope this helps.