In terms of big picture stuff, you're absolutely right that many DNNs are more "inspired by" the brain and less a faithful model of it. However, a lot of the things mentioned in your post are either overstated or outright wrong. For example:

1. Neurons, or more specifically, connections between neurons ("synapses"), absolutely do have weights, and the "strength" of synapses can be adjusted by a variety of properties that act on a scale of seconds to hours or days. At the "semi-permanent" end of the spectrum, the location of a synapse matters a lot: input arriving far from the cell body has much less influence the cell's spiking. The number (and location?) of receptors on the cell surface can also affect the relative impact of a given input. Receptors can be trafficked to/from the membrane (a fairly slow process), or switched on and off more rapidly by intracellular processes. You may want to read up on long-term potentiation/depression (LTP/LTD), which are activity-dependent changes in synaptic strength. There are a whole host of these processes, and even some (limited) evidence that the electric fields generated by nearby neurons can "ephaptically" affect nearby neurons, even without making direct contact, which would allow for millisecond-scale changes.

2. While you can start by dividing neurons in excitatory and inhibitory populations, there's a lot more going on. On the glutamate (excitatory) side, AMPA receptors let glutamate rapidly excite a cell and make it more likely to fire. However, it also controls NMDA channels that, under certain circumstances, allow calcium into a cell. These calcium ions are involved in all sorts of signaling cascades (and are involved--we think--in tuning synaptic weights). GABA typically hyperpolarizes cells (i.e., makes them less likely to fire) and is secreted by cells called interneurons . However, there's a huge diversity of interneurons. Some seem to "subtract" from excitatory activity, others can affect it more strongly in a divisive sort of way or even cancel it completely. Furthermore, there's a whole host of other neurotransmitters. Dopamine, which is heavily involved in reward, can have excitatory or inhibitory effects, depending on whether it activates D1 or D2 receptors

3. While the textbook feed-forward neural networks certainly have "instant" signal propagation, there are lots of other computational models that do include time. Time-delay neural networks are essentially convnets extended over time instead of space. Reservoir computing methods like liquid state machines also handle time, but in a much more complicated way.

4. I chuckled at the idea of finding a biological correlate analog for reinforcement learning, since reinforcement learning was initially inspired by the idea of reinforcement in psychology/animal behavior. People have shown that brain areas--and individual neurons within them--encode action values, state estimates, and other building blocks of reinforcement learning. Clearly, we have a lot to discover still, but the general idea isn't at all implausible.

Finally, some people are fairly skeptical that the fields have much to learn from each other; Jürgen Schmidhuber said this a lot at NIPS last year. However, other, equally-smart people (e.g., Geoff Hinton) seem to think that there may be a common mechanism, or at least a useful source of inspiration there. But, if you want to work on something like this (and it is awesomely interesting), it really helps to have a solid grounding in both.