There are a lot of questions here, and I'm more sure about the answers to some of them than I am about others. But nevertheless, I'll give it a go.
The first and most important question is what would the temperatures be like. This is the one I'm least sure about. The reason I'm unsure is that it's highly dependent on a lot of things that would probably be quite different on a tidally locked Earth. These include the greenhouse effect (which is highly dependent on water vapour), albedo (which depends on how many clouds there are, as well as on ice), and most importantly on the wind speeds.
If there was no heat transport from the hot side to the cold side then we could expect the temperatures to be similar to the temperature range on the Moon - around -150 °C on the cold side, and more than 100 °C on the hot side. However, if the planet has an atmosphere then it will transport heat from the hot side to the cold side, and this might make the temperature difference much more moderate. This could only work if the rate of heat transport (and hence the wind speeds) were quite substantial, because if it gets too cold on the cold side then the atmosphere will start to freeze. This will leave less air to transport heat, leading to more freezing, and so on in a feedback loop that would result in an airless world. This is probably the most likely outcome of a tidally locked Earth-like planet - but it's not so interesting from the point of view of fiction, so I'll focus on how an inhabitable world might be plausible.
A high enough heat transfer rate might be possible if the hot side were very hot, perhaps because of a large greenhouse effect in addition to permanently facing the Sun. I'm not sure whether this would mean the hot side would have to be so hot that the ocean would boil, but I suspect so. In this case we would expect to find most of the water in the form of ice on the cold side - but perhaps some of it could be found in a liquid state near the boundary. One would hope there could be some kind of water cycle - I'm imagining something like glaciers being continually melted by the warm air blowing in from the hot side, with the meltwater flowing in huge rivers (or even flowing seas) to the hot side, where it evaporates and cycles back around to fall as snow on the cold side.
Next we need to consider what the air flow patterns would be like. The situation on the present-day Earth is complicated, but, oversimplifying a bit, the basic principle is that air heats up at the equator, then rises, then travels towards the poles at high altitudes, then cools (due to emitting thermal radiation into space), then sinks again and travels back toward the equator. So the prevailing winds at ground level tend to blow towards the equator. On a tidally locked Earth I'd expect a more extreme version of this phenomenon. The flow patters in the middle of the hot side would probably be crazily chaotic - there'd be a lot of energy there to drive storms, especially if there was a water cycle providing moisture - but in the region near the boundary I would expect a fairly stable convection cell, with hot air blowing to the cold side at high altitude, sinking, and blowing back to the hot side at low altitude. So a person standing on the surface near the boundary would experience very strong prevailing winds towards the sunlit side. Exactly how strong these winds would be, I can't say, and it's also very hard to say how much cloud cover or rainfall there would be. However, in general a higher temperature difference means more energy goes into weather systems, and on this world the temperature difference would be bigger than on Earth, so we'd generally expect the atmosphere's dynamics to be stronger and more violent.
Something to note is that heat can be transferred as latent heat rather than simply hotter and cooler air. Water absorbs heat when it evaporates and releases it when it condenses. So if there is a strong water cycle, with lots of cool liquid water flowing towards the hot side and lots of warm water vapour being transported by the atmosphere toward the cold side, then you can have a higher heat transport without needing such strong winds. In this scenario, someone on the surface would experience a prevailing wind towards the sunlit side, but looking up at the sky they would see clouds moving in the opposite direction.
An extra complication on Earth is the Coriolis force, which causes the prevailing surface winds to blow around the Earth instead of just from pole to equator, and also allows hurricanes to form. It also causes the jet streams. But a tidally locked Earth would hardly be rotating, so there wouldn't be a Coriolis force and these phenomena wouldn't happen - the prevailing winds near the boundary would blow directly towards the Sun, and rotating hurricane-type storm systems would be extremely rare if they existed at all.
The next question is about whether there would be a distinct Northern and Southern ocean. I don't think this would be the case. This idea comes from the fact that the Earth's rotation causes the oceans to bulge out around the equator - but it also causes all the rock to bulge out, too, which is why we don't have just a big ocean around the equator. I guess you can imagine that if the Earth somehow stopped spinning then the oceans would flow towards the poles, but the rock would move much more slowly, so that there would be a period of time where there would be oceans near the poles but land around the equator. But over longer geological time scales the rocky part of the Earth would change its shape as well, and you'd end up with roughly evenly distributed oceans again.
The same goes for the question about the Sun causing a permanent tidal bulge in the oceans. It would, but it would also bulge the shape of the planet, so you wouldn't necessarily have all the ocean on the sides facing toward and away from the Sun. I think the fact that the water would tend to freeze on the cold side would have a much bigger effect on water distribution than anything else. Of course, the Moon would still cause tides, if there was a moon.
Next is the magnetic field - I don't feel qualified to answer this one. I've always wanted to understand how Earth's magnetic field is generated, but I've never found a good explanation, so I don't know how plausible it is to imagine it working without the Earth's rotation. The importance of the magnetic field is that prevents the atmosphere from being slowly blown away by the solar wind, so it would be needed for this planet's atmosphere to persist in the long term.
I'm also not sure how to answer the question about the Moon. It seems hard to imagine how the Moon could be in orbit yet the Earth still be tidally locked to the Sun, but I'm no expert.
Another potential big difference is plate tectonics. If the oceans are frozen on the cold side then this will put a lot of weight on the plates there, which will probably change the dynamics of the whole system - but unfortunately I'm not currently able to imagine what the result would be. Plate tectonics are quite important for recycling elements over very long time scales. The water cycle on this planet would tend to transport nutrients to the hot side and deposit them there, so you'd need geological activity to recycle them in the long term.
One more really important thing we have to think about is photosynthesis. It will be very difficult for plants to persist on this planet, because the hot side will be very very hot, and the cold side has no light. So probably there would be much less life on this planet than there is on Earth, with most of it existing towards the edges of the hot side. (The prevailing winds and water currents here will be from the cold side, so these could keep things relatively cool even though the sun is always in the sky.) If the cloud cover is 100% then this will also make photosynthesis harder due to reducing the available light. This is important because it might mean there wouldn't be enough oxygen to support human life, so I guess in designing this fictional world there is a tradeoff between having a strong water cycle to transport lots of heat, versus having it weak enough that enough light reaches parts of the surface.
You mention the possibility of life on the dark side. This isn't impossible of course, but you'd have to consider the question of what it would eat. Most of the life on the sea floor, for example, eats goo composed of dead stuff that drips down from the surface - so its power source is ultimately sunlight, which is converted into chemical free energy by photosynthetic plankton. So while there could be some life on the dark side (feeding off geothermal energy from hydrothermal vents under the ice, for example) there couldn't be very much of an ecosystem there unless there was a constant source of chemical energy being transported somehow from the light side. But it's hard to imagine how this could happen without making the composition of the atmosphere very different from Earth's.
I hope this provides some food for thought. It is of course all very speculative, and there may well be important things that I haven't thought of, or where I've got something wrong due to lack of expertise - so caveat emptor.