I'll deal with the questions in reverse order, because your later questions are easier to answer.
If the oceans froze several feet deep (let's make the math easy and call it a meter - about 3.3 feet, fits the criteria), it would have no effect on tectonic plate movement. Let's think about the Americas moving west, into the Pacific Ocean. The Pacific Ocean is 165.2 trillion square meters, so we have 165.2 trillion cubic meters of ice in the way of the continents. This means we have a mass of about 1.5 quadrillion (1.5 * 10^15) kg of ice (note: I used the density of pure ice because the density of salt water ice depends on the level of salinity, so could change place to place). Which seems like a lot. But a ballpark estimate of the mass of a tectonic plate is on the order of 40 * 10^22 kg (http://www.answers.com/Q/How_much_does_a_tectonic_plate_weigh <-- not the greatest source, I know, but we're talking estimates here, and the math checks out). So the plates are still over a million times more massive than the ice. They're not slowing down.
As far as rivers are concerned, they'll freeze too. The oceans have a lot of water in them, and all of that water has thermal energy. As rivers have much less volume than oceans, they'll freeze way before the oceans do. How it would affect the overall ecosystem, I'll get to that in a second. Spoiler alert: it won't be good.
So, the million dollar question. Can it actually happen. Yes, though it's not much fun for anyone involved. It's similar to the Great Lakes freezing over (which has happened for individual lakes, but not the system as a whole - the record for the whole system is about 95.5% ice cover). You'd basically need temperatures to go below freezing, globally, for long enough that the water in the deeper parts of the oceans stop effectively warming the surface. Once heat transfer in the oceans slows down, the surface will start freezing (note: the coasts will freeze before the middle because of this, as there is less water near the coasts than in the middle of the ocean). Unfortunately, going back to the effect on the ecosystem, the temperature drop will freeze a lot of land water as well - not just the rivers, the water in dirt and such. So if it went on that long, the entire ecosystem would start to collapse - flora can't get enough water to grow, fauna can't get enough water to avoid dehydration, and eventually the food chain crumbles. And that's not even talking about marine life that suddenly has to get used to life in way colder, darker conditions. It won't be pretty.