Does my city need all three zones (industrial, residential, commerce) or can I spread them out between cities?

Can I build all industrial zones in one city and then put all my residential and commerce zones in another city? Would this still provide jobs for my sims or do I have to build all three zones in the same city?

Also wondering if water pumps provide water, as the last Sim City I played they had to be placed near a river?

You can indeed specialize your cities such that each of them only contain one type of zoning, as long as your cities are connected at the borders with roads and other types of transport. It does, however, come at a penalty to commute time, as your residents have travel to the city border before entering the next city to work.

It is important to note that while Sims can cross borders, pollution cannot, so isolating pollution in this manner is effective. Commercial buildings do not produce pollution, so they can be build next to residential areas with no ill effect, and thus there is less benefit to zoning them in a separate city. However, as mentioned in the comments, crime does decrease with residential population, and as crime affect commercial building strongly, there are some benefits to isolating commercial zones in a separate city. However, the increase in commute time in my opinion mostly negates this benefit.

Also note that while you're in one city, time (and therefore, RCI demand and neighbour deals) is frozen in all others, thus it is a good idea to switch between cities periodically if you are doing this, otherwise there might not be enough demand or supply of utilities for one city to grow on its own.

In general it is good to have multiple cities connected in a region - SimCity 4 is designed to allow that form of connectivity, and the game plays better with more cities in the region. However, building only one type of zones in a city is a little extreme, and some of the things mentioned above could be seen as cheating since they exploit the somewhat unrealistic limitations of SC4's simulation engine. Whether you chose to exploit these 'features' is entirely up to you, since SC4 is a sandbox game, and there's no win state.

You can build the different zones in different cities as long as you have a good transport network in place along with the connections between the cities.

Water pumps don't need to be near a river, they just need to be connected up the water pipe system, but keep them away from pollution as they will affect their effectiveness.

Also I would suggest looking at the following sites as they have tons of information (and mods) for SC4; Simtropolis and SC4 Devotion.