Hot days and hot nights implies to me that much of the planet is in a humid environment with a strong greenhouse effect. Normally, dry air and soil heated by the sun during the day will lose most of that heat back out to space during night, but this loss can be slowed by a stronger greenhouse effect or higher local humidity (water vapor is great at holding heat, and when it condenses back into dew it even releases heat!). One good way to explain a worldwide high humidity would be having much smaller continents than we do on Earth, say by scattered islands in a large ocean (expect strong ground-level winds at nightfall and in the morning, in this case). The best explanation for stable warm temperatures is regular intense sunlight, but temperature could also be increased by, say, increased geothermal activity, or tidal heating (does that work with a flat world? who knows!). If we assume that sunlight during the summer day is quite intense (but with about the same amount of UV light that we get on Earth), plants with access to water should be well adapted for rapid photosynthesis.
During the winter, trees and other plants would be able to photosynthesize only for the portion of the time that their tissues were above freezing. Add in that the best explanation for lower temperatures is reduced solar radiation (and not something weird, like winds blowing off into the void at the edge of the map only during the winter), and you have a situation where plant growth is stifled both by temperature and by solar radiation. Certain organisms on Earth are adapted to one or the other (diatoms can continue to photosynthesize below -4 degrees Celsius, and plants in the rainforest understory can survive with just a few % of full sunlight intensity), but maintaining the adaptations for photosynthesizing in both freezing conditions and low light may be too costly for many organisms to adapt for it. Compare plants in Earth's arctic and subarctic regions - while it's theoretically possible for them to adapt to photosynthesize during part of the winter period, most plants in these regions are active during the summer.
The coastal waters are probably as full of life as they are on Earth, but you might expect vast areas with low photosynthetic production in the centers of oceans, given there's no Coriolis effect nor north-south temperature disparity to drive currents and redistribute nutrients. You could still see some ocean currents as the seasons change, driven by the land masses being warmer (at the end of summer) or colder (at the end of winter) than the water.
For the climate scheme I propose, with humid summers with high solar radiation and cold winters with low insolation, you probably see something similar to the temperate coniferous/leafy mixed forests of Earth, swapped out for plains on highlands and in soils with low nutrient availability. Areas sheltered from the sea by mountain ranges would look much like Earth's deserts, but there probably aren't too many of these if we assume the land masses are small and humidity is high. Precipitation is likely to be evenly distributed along most coastlines, without the prevailing winds that we get from Earth's rotation and without a disparity in sunlight amount between a northern and southern hemisphere.
Most prey species probably either go into a mode of low activity during one of the seasons, or change their coat colors to be better able to hide in the snow/vegetation. Prey are probably mostly nocturnal (given that the summer is the best time to acquire food, and the summer nights are so long), though this could change in environments with high visibility where being able to outrun a predator is a better bet than being able to hide from them. It'd be hard to determine much more about the animal life in this world from the seasonal scheme, but anything that exists on Earth could probably exist in this world, as well.