Yes when referencing the negative temperature coefficient, that’s the water (and in general the heat of the whole core) which lowers reactivity due to inefficient neutron-slowing. The effect is even more pronounced of the water is allowed to boil to steam. The steam bubbles making “voids” which won’t slow neutrons, basically at all.

Neat trick can be played here. Thermal output can be controlled in a BWR reactor by increasing or decreasing the coolant flow. Faster flow will drive the threshold of boiling water higher, allowing more of the fuel rod to fission. If they want to slow the reactor, slow the coolant flow rate. The water will boil lower and reaction rate slows at the top of bundles. BWR control rods insert from the bottom so it can control reaction rates from bottom up and top down.

Not really. The uranium atom would be in the ground state. Higher temperature would increase the KE of the U235 atom but it would also increase the average KE of neutrons available. And as mentioned elsewhere, higher KE neutrons have less probability of being absorbed.

So for cores designed for thermal neutrons they have a “negative temperature coefficient” or if the reactor gets hotter, fission rate decreases, bringing the temp back down. This is a nice feature to keep the reactor controlled, but it wont prevent a meltdown outright.