How can I keep extra virgin olive oil cool and usable, while it's hot in the summer and when the fridge solidifies the oil?

As @GdD said in his comment - you can't, unless you have a way to cool it to a defined temperature such as a cooling water-bath. Olive oil is perfectly fine at higher temperatures than 25 - this would be on the cool side for a Greek (as big users and preparers of olive oil) summer, especially before the presence of air-conditioning:

Although summer in Athens began in June, on average the weather reaches its highest temperatures in July. The average temperature this month is 27°C (81°F), while the average low is a warm 22°C (72°F), and the average high is a stifling 32°C (90°F).

From here.

As to slowing down oxidation from the free-radicals, yes, keeping it in the fridge will work to some extent as in the crystalline phase the oils will be less susceptible to oxidation. Indeed I found one study that looked at oxidation and storage of olive oils at different temperatures and with different packaging types, who found that the shelf-life of the oils was best in green glass (GG) and at 6 Celsius - see quote below (EVOO is Extra virgin olive oil, TT is tinplate tin).

While shelf-life of EVOO was differently affected by packaging and storage temperature, the latter being critical for the oxidative changes taking place in oil, at the end of the observation period none of the oil samples showed significant changes in the visual descriptors of clearness, green and yellow reflections, and the basic positive sensory notes of bitterness and pungency were maintained. In particular, the oil stored in GG at 6 °C mostly preserved positive attributes, whereas the one stored in TT at 26 °C showed an enhancement of oxidative processes leading to a significant presence of the rancid flavour. Moreover, GG6 maintained the highest BI and did not show defects at the end of storage, further suggesting that storage in GG at a low temperature (6 °C), could represent a promising storage condition to slow-down the oil degradation during market storage.

Incidentally they found that storage under any conditions for more than a month meant that the oils no longer met the legal requirements for extra virgin oil. This would imply that your best bet for keeping oil is indeed to store it in the fridge and in a green glass bottle, and use it rapidly.

However, as you noted, storage in the fridge results in the oil becoming solid, so you either have to live with some solid oil and re-warm it each time you want to use it, or come to some compromise such as just storing it in the dark but not cold.

Not all refrigerators have temperatures that will coagulate olive oil. The best way to preserve the integrity of the olive oil is in a wine cooler set at 12-15°C. The olive oil does not coagulate or harden.