How to calculate required current and specifications for a "wet coil" to generate x Tesla of magnetic field?

Here is an approximation of what you are trying to do. The magnetic field inside a solenoid is given by (from Wikipedia) $B={\mu}nI$ where ${\mu}$ is the permeability of the medium (presumably water in your case), $n$ is the number of turns per unit axial length, and $I$ is the current. The maximum value of $n$ is determined by the thickness of the wire you are using. This vendor sells magnet wire as thick as 12 gauge, which for a single layer wrapping yields $n=468$. The permeability of water is very close to that of vacuum, so use ${\mu}=4{\pi}{\cdot}10^{-7}$ $H{\cdot}m^{-1}$.

Achieving your minimum desired field of 0.1 Tesla requires a 170 Amp current which is probably well beyond the safe operating range for 12 gauge wire. Now this is an engineering problem. You could look for a thicker wire and also wrap multiple layers to reduce the required current. If the water is flowing that will help keep the wire cool, but being submerged demands careful consideration of electrical isolation. Here are some epoxies for potting electronics, though I suspect there are many other options available. Hopefully the links provided help illuminate what keywords to use in your research.

Once you have an idea of what kind of wire to use (and its resistance), and also the max current you'd like to run, you can use $P=I^2R$ to estimate the power you will need. Then you can start shopping for a DC power supply.

A 1 Tesla field is huge!

To the senior members: I apologize in advance if it was in bad taste to link to vendors.