A very comprehensive summary,
@Ridgerunner , you stole much of my fire.
Once certain mulches rot they add a lot of nutrients to the soil. To me this is a huge benefit of certain mulches.
For that reason, my preferred mulch is green-cut hay, not straw. The hay releases some nutrients as it breaks down - and breaks down quickly. It also is a good food supply for the worm population. By the end of the summer, worms will have consumed at least 1/2 of the hay; but at that point, weed suppression is no longer an issue. I may lay additional hay or grass clippings in pathways, enough to prevent mud splash & keep me from tracking mud into the house.
Personally, I don't like straw as mulch. It is effective as a weed barrier, but IMO is
too effective at blocking the sun's warmth. It also breaks down much more slowly if turned under, and some level of wheat seed - sometimes excessive - is unavoidable. Hay can introduce weed seed too, and I've had a few bad lots over the years... but overall, my results using timothy, clover, or alfalfa hay have been very positive.
Straw or wood chips and stuff like that gives some bugs like squash bugs great hiding places. I've seen the recommendation to not put mulch right up to the plant but leave a bare spot around the stem to protect them from critters. You can't see mole or vole tunnels. Some diseases live in the ground like blight, the spores can splash up on the plant in a rain or if you sprinkle water. Mulch can help stop that. Mulch right up next to the plant may keep it wet so certain molds can grow. It's a mixed bag.
As a rule, I will not apply hay or any organic mulch around vining crops, because that just provides habitat for squash bugs and cucumber beetles, and makes it harder to eliminate them when they appear. Plastic or agricultural weed cloth would seem to be better under the vines, but they can also be an invitation for voles to move in (as I found out last year). Protected from prowling cats, voles began raising their young in my gherkin patch last year, emerging from cover just long enough to chew off half of the vines at the ground.

Once exposed, and after I quickly dispatched their young, the voles vanished.
Black plastic can be one of the keys to success when growing heat lovers in cooler climates though. Sweet potatoes and watermelon in particular will benefit from the higher soil temperatures... but I will try to use only a foot or two around the plants, in hope that the smaller covered area will allow the cats to keep the voles at bay.
In my garden, weed control is issue #2 - the primary purpose of mulch is to prevent the negative effects of mud splash. Beans in particular really benefit from that. I don't apply mulch around seedlings, since that can encourage slugs & bugs to attack the plants while they are most vulnerable. So most years, I don't mulch the beans until they have at least 1-2 true leaves. The seedlings typically look somewhat chlorotic & unhealthy at that point; but after being mulched & sprayed down to remove soil from the leaves, they turn green with a few days. The transformation is sometimes quite dramatic. The stabilization of soil moisture also improves blossom set & pod quality. The improvement of pod quality is especially notable for yardlong beans.
Mulching around tomatoes & peppers helps to stabilize soil moisture, which in turn improves blossom set & reduces blossom end rot. The results with peppers honestly surprise me, since I would have assumed that lower soil temperatures would reduce production, and that doesn't seem to be the case. I remember reading somewhere that peppers benefit from higher humidity, and perhaps the moist hay helps to maintain higher humidity around the plants. I am also surprised that the eggplants I now grow prosper when heavily mulched, since eggplant is as heat-loving as okra... but based upon my early trials (where most eggplants died shortly after flowering) most eggplant varieties would probably not have responded as favorably.
A heavy mulch of hay does offer potential habitat for mice & voles; but that doesn't seem to be the main problem, since hay offers no protection from cats. I find that mice & voles seem to be attracted less to mulch, than by the dense leaf canopy that develops in mid season.
I DO pull seedless weeds and throw them on my lawn and use my mower to "compost" them. Just sayin...
I throw any weeds
with seeds onto my lawn, for the same purpose. Few garden weeds can compete with sod, and those seeds will never find their way back into the garden. All other weeds are either left to die in the garden where they were pulled, or dried & used elsewhere as mulch. The only exceptions would be things like purslane, crabgrass, or anything else which is likely to re-root.
Regarding compost: like all of the farmers locally, I compost in situ. After the frost puts an end to the garden, I remove all poles & hardware, then mow the entire garden until everything is finely chopped. This encourages the rapid decomposition of plant debris. When weather allows, I will turn everything under, with the addition of shredded leaves whenever possible. Any trimmings or peelings taken out of the garden are returned to the garden... I do not introduce foreign material into the garden from supermarket produce, to avoid the possible introduction of new diseases. Those materials get tossed into my tree line, where scavengers generally destroy them quickly.