What is the temperature of the flowing water in icy river?
This is a complex question.

I will assume the following to be able to frame an answer properly:


Let's take a general cold area as a baseline, such as the High Arctic or 70°+ N latitude;
The reference for this example can be Pond Inlet, on Baffin Island, an example where long-term records are available (72° N). The following data assumptions will be based on the normals provided by this station. But what is discussed is likely to be valid for a large area of the High Arctic - like 70°+ N latitude;
In this area, the normal (1981-2010) mean annual air temperature is -14.6° C is fairly representative
Also, in this area, the normal yearly degree days to reach thawing (TDD - the sum of daily average temperatures above 0° C) is 473 degrees.
The bulk of the TDD sum (473 degrees) occurs during June, July and August; only 2 degrees in May, 31 in September and 1 in October, so thawing is most likely minimal outside these three months. As a consequence there are two fairly distinct seasons: the thaw season (summer, duration 3 months) and the freezing season (winter, duration 9 months).
Pro-glacial rivers are not considered (river taking their source from glaciers, different story)
Lakes are not considered (different story)


This all means that the cold region's rivers are not thermally static through time. Even in the High Arctic, an icy river can be frozen along both its vertical and horizontal profile during a good part of the long winter, and thaw completely during the short summer. But other considerations exists.

Let's examine what can happen when the snow thaw is sometime in June. The discharge signature for a river in the High Arctic is nival, with a discharge peak occurring during early summer due to the massive input of snowmelt water, and additional weaker responses from the small rain amounts. This looks like this:


Source:  Woo (2012) Permafrost Hydrology, fig 10.14; from river McMaster, 74° N.

So what basically is happening here?


Peak is during June.
Afterward, once the snow had melted completely, the discharge lowered drastically.
By late August, there is almost no more water in the river, and it may be completely recessed.
Then winter arrives.


During the next several months, snow will accumulate in the landscape. In a landscape there are snow sources and snow sinks (Pomeroy et al. 1997);  sources may be higher lands, outcrops, plateaus, or exposed areas, and sinks lower areas, such as depression or channels. Our typical nival river will be a sink in this context.

Because of this, it is not only patches of ice that will be found in the river channel, but tightly packed snow that builds up at the base when it is a thick layer (or is looser when the snowpack is thin). In any case snow will be most likely metamorphized in structure as a result of surface thaw and the water snowmelt infiltration -> snow percolation -> re-ice events during the winter.

What is happening in June, in more detail?
Let's analyse this further with a more detailed (though raw and unprocessed) graph from a small stream in a gully near Pond Inlet, with resolution available up to the nearest day:


Source: Stream temperature and water pressure in a stream located in a gully on Bylot Island, 85 km from Pond Inlet; raw, partial data from my PhD work. Temperature in blue, scale on the right, and pressure in black, scale on the left. Dates on the x-axis.

By looking at the pressure line (in black), it can be interpreted that there was no water until June 16th when the pressure increased drastically. Temperature (in blue) stayed at -2 °C until the 17th. Water was infiltrating into the snow at this time, contributing to the slowly rising pressure, but the water did not reach the sensor until the 17th when the temperature starts rising. During the 18th, temperature rises to -0.5 °C, then hovers in these negative temperature for almost 24 hours. This supercooled water temperature is due to a mix of factors, such as increased pressure from the water on snow, the release of latent heat due to the metamorphized snow thawing, and the near zero temperature of the water, in balance with the snow. 

When a critical mass of water accumulates near the logger emplacement in the stream channel, the bulk of the snow melt and water then shifts to positive temperatures, still near zero, with diurnal/nocturnal shifts (blue line becoming wave-like).

Note: This small stream is minimally impacted by the geothermal flux (heat flowing up from the ground below).

Summary, in relation to the questions:


Temperature will be mostly above zero; but may be under and near zero for short periods when the water is mixing with snow and ice in a channel;
Yes - as shown in the second figure, water was at -0.5 °C for about 24 hours;
I think it is common, as thawing temperatures do reach very high latitudes such as Ward Hunt Island (83° N), and thus this process likely concerns a large area, including probably the entire Canadian Archipelago (lowlands at least).  But this transition process does not last long at any location. Additionally, it is worth noting that High Arctic rivers are difficult to access, monitor, and study, so it is difficult to generalize this with confidence; other factors or contexts could exist where conclusions may differ.