Yes this seems to be true for now. However keep in mind, that the hot months are just beginning and have a high impact on the energy production of France.

Here are charts I found. They are a bit difficult to read: Basically the width of the connection on a country determines the amount of energy they sent over the connection. So the connection from France to Switzerland which is much broader on the FR side says France exports more than it imports from Switzerland. For Spains the reverse is true as the connection is broader on the ES side.

To see the impact of hot months compare the charts for May and July-September of last year.

I think France will in average stay a net exporter for the next years, but in the trend they will shift to more importing as many of their nuclear reactors need maintenance or replacement and all of them are affected by climate change. The trend may of course change again due to changes in politics.

Not really

~~That is true.~~ [Édith: apparently not!‽]

However, import/export stats usually just show where energy is cheapest. Of course it's cheaper to use French excess nuclear energy than start a German gas/coal plant. Germany has more than enough (fossil) capacity on its own, however.

The 2022 situation in France is an outlier in this regard obviously, because it temporarily reduced French generation capacity below their needs. Also outliers: Countries like Italy and Hungary that never had enough generation capacity in the first place and do need to import rather than importing to reduce cost.

I thought that if it came to an emergency they could keep the reactors running, only the exhaust water would be too hot and river wildlife would die around the vents.

France has already upped the maximum temperature legally allowed two or so years ago. And while short term that is a solution, the fact is that us humans need working ecosystems to survive. It's not just fishers being impacted, ecosystems are connected and we're cutting into them at record pace anyway.

So instead of running hot, the river runs dry?

Any thermal electricity plant uses an insane amount of water. For every kWh of electricity generated, about 1.5-2 kWh need to be cooled away.

A pure air cooling system is theoretically possible, but it drastically reduces the plants efficency, as the energy you can recoup at the turbine is directly dependant on the temperature difference between the hot and the cold side.

So in any way thermal plants are never going to be an option that is favourable to build now, over building renewables, except for a small degree of net stability that can be provided by already existing plants.