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For a civilization that depends heavily on canals for transportation, what limits are there on how steep canals can before they flow too fast to traverse? I know that barges were normally pulled by animals upriver, if winds were unfavorable, so would the maximum slope of the canal depend on how strong the beasts of burden are? It would be neat to be able to calculate the slope in like meters dropped per kilometer in a general sense.

A canal with a gradient and a flow is a river. And navigatable rivers are pretty much flat. Taking a nearby river (the Medway) and measuring the change in height from the coast to the point at which it becomes essentially unnavigatable gives an average gradient of 0.0003 or about 30cm per kilometer. Wider rivers have lower gradients. The Mississippi has a gradient of about 1cm per km, yet still flows pretty quickly

If you have a flowing canal, you need a constant source of water. This is why canals use locks. Don't design canals as rivers, unless you already have a river to redirect through it.

The Suez Canal is an artificial waterway of about 190 km in length running across the Isthmus of Suez in northeastern Egypt which connects the Mediterranean Seas with the Gulf of Suez, an arm of the Red Sea. It has no locks, because the Mediterranean Sea and the Gulf of Suez have roughly the same water level and is thus the world's longest canal without locks. (Source)

Karakum Canal without locks in Russia is the longest canal in the world with a total length of 1400 kilometer. It was built in 1954 and completed in 1967 having a length of 840 kilometers and further extended in 1970 and 80s up to Caspian Sea coast making its total length of 1400 kilometers. (Source)

To answer your question, 0°.

The differences in "sea level altitude" between the sources and destinations of these bodies varies naturally according to the tide, but the canals are dug such that those variations are accommodated within the canal — which can only be done when the source and destination altitudes are fundamentally equal.

There is obviously some small amount of ascent/descent or the canals/rivers used for cargo transport wouldn't have a current. But that ascent/descent need not (and almost always is not) a condition of the path being traveled by the cargo ships. For example, the Mississippi has nearly all of its descent in its northern most reaches. By the time you get to the majority of shipping in the southern U.S., it's basically flat. The current is caused by water pressure from the north.

The gradient should be 0, UNLESS the high end of the canal has a flow rate high enough to compensate for the amount of water being drained from it by the canal into the low end.

Which for any decent sized canal would be a lot indeed and probably make the upstream end of the canal hazardous for shipping because of the rapids.

This is simple physics. Water (as any liquid really) want to establish equilibrium, meaning the surface wants to be flat and level. Thus it flows from high to low until equilibrium is established, both vessels having the same water level.

The only way to thus have a canal with one end higher than the other without an artificial barrier (a lock) in it is to ensure that the high end gains water as fast as the canal can drain it.

In theory of course you could calculate the exact drainage and pump water out of the low end at that rate, back into the high end. But locks would be much easier, cheaper, and more reliable.

Your only limiter is how much your beasts of burden can drag up river. This is less dependent on slope than it is on the speed of flow and shape of the hull. Personally I'd favour double hulled boats if your channel is wide enough, they cut through water pretty well while giving you width for cargo space. Whereas a similar single hull would probably need to be a long thin boat in a heavy current.