Let me lead off with an excellent book on the topic: Corky Bell's Maximum Boost. There is a sound treatment of the basics of turbocharger operation in addition to some dated and esoteric applications that are still interesting. For example, I find the discussion of turbocharging different types of carburetor to be of intellectual if not practical interest.
To summarize the points called out in your question, here are some of the major aspects of the turbocharged engine that are of interest:
The air flow: remember that an internal combustion engine is effectively an air pump. If we're talking about an engine that is operating "under load", we can assume that you've opened the throttle. For example, when rolling down hill, you don't need to hit the throttle so the entire intake to exhaust path is pumping a smaller air mass. However, driving uphill will require you to open the throttle (give it gas), adding air to the intake. This causes the engine computer to add fuel to the mixture. The fuel-air mixture is burned to produce engergy. The exhaust from this combustion then proceeds to ...
The turbine: this is part that looks like the front of a jet engine sitting in the exhaust gas path. The turbine sits on one end of a spinning shaft. On the other side is the compressor. That is the part that actually makes the boost on the intake side of the engine. The more exhaust gas pushes past the turbine, the more it wants to spin and make boost on the compressor side. However, there is also ...
The wastegate: this is a valve that also sits in the exhaust gas path. It provides a short cut for the exhaust if the engine does not really need boost at this time. This can be used for peak boost control (too much boost can physically destroy your engine). This can be a purely mechanical sprung valve that stays closed up to a certain positive pressure in the intake path and then progressively opens as boost increases. It could also be under the direct control of the engine computer. For example, my car (in stock tune) was very annoying in its refusal to stay at peak boost in third gear. It also would refuse to boost past a certain point with partial throttle. The engine computer was effectively saying "no, that's enough fun for now."
For example, if I am rolling downhill in gear with my foot off the gas, the throttle is closed. There isn't enough air mass passing through the intake to exhaust path to make the turbo spin, wastegate or not.
However, the scene changes at the bottom of the hill as we climb the next rise. I have to open the throttle to get up the hill. If I'm in low gear, the RPMs will be higher, the exhaust gas energy will be higher and the turbine will spin right up. However, since I'd require partial throttle at lower gear for the same acceleration, my engine computer might veto boost past a certain point, opening the wastegate.
If I'm in high gear, the RPMs will be lower and I'll have to open the throttle wide open to get up the hill. However, the exhaust gas volume and velocity will be low and it's possible that I won't have enough energy for the turbo to make any significant positive pressure (e.g., around 40 mph in fifth in my car). Even though I'd really like to make boost in this situation, I won't be able to.