Plants create oxygen as a byproduct of photosynthesis, using energy from sunlight to break apart molecules of water and carbon dioxide and reform them into carbohydrates (such as glucose) and oxygen. If you want a substantial amount of oxygen in your atmosphere, that oxygen has to come from somewhere- either from oxides in the ground (most likely quartz crystals or metal ores), or from CO2 in the air. And liberating oxygen from any such source requires energy. That energy has to come from somewhere.
There are a few exothermic reactions that release oxygen (such as the "oxygen candles" popular on submarines: $3NaClO_3 + 2Fe \to 3NaCl + Fe_2O_3 + 3O_2$), but their reactants (sodium chlorate and iron metal, in this case) have to be more reactive than oxygen in order for the reaction to work at all. By the time life starts developing on your planet, any such compounds will have already broken down, and any oxygen released will be bound up in things like quartz, iron oxide, and carbon dioxide.
You mention fungi and the parasitic dodder plant, and that they do not produce oxygen. This is true. Fungi are actually more closely related to animals than to plants; they produce energy by breaking down biomass produced by other organisms and reacting it with oxygen to produce water and carbon dioxide.
In order to produce oxygen, your "plants" will need to need another source of energy to break down oxygen-containing compounds (most likely carbon dioxide) in order to get at the non-oxygen components, and release the oxygen as a byproduct. All Earth plants power this process with sunlight, because it's abundant and it works. However, some organisms have other energy sources. The chemosynthetic bacteria in giant tube worm guts, for instance, create carbohydrates by reacting carbon dioxide with hydrogen sulfide:
$$12H_2S + 6CO_2 \to C_6H_{12}O_6 + 6H_2O + 12S$$
This produces solid sulfur as a byproduct, rather than oxygen; however, a chemosynthetic organism with a source of hydrogen could react it with carbon dioxide instead:
$$6H_2 + 6CO_2 \to C_6H_{12}O_6 + 3O_2$$
Such a hydrogen-eating organism might release oxygen as a byproduct; or, more likely, it could react the oxygen with more hydrogen to produce water and quite a bit more energy.
$$12H_2 + 6CO_2 \to C_6H_{12}O_6 + 6H_2O$$
However, I doubt that chemosynthesis will play a major part in your planet's ecosystem. As on Earth, chemosynthetic organisms will be relegated to dark, oxygen-poor areas with sources of reactive chemicals such as hydrothermal vents at the bottom of the oceans. Instead, the foundation of the ecosystem will be based on photosynthesis.
You've got a planet with a high axial tilt where, even in the dead of winter in the polar regions that get only a few hours of sunlight per day, the temperatures rarely drop below freezing. There's only two ways for that to be possible: either there's some serious greenhouse effects going on, or the lower latitudes and the summer months get a ridiculous amount of sunlight. If there's a strong greenhouse effect, that means that there's a lot of carbon dioxide, methane, or other greenhouse gases in the atmosphere. Gases such as methane will quickly react with all the oxygen that you have; while the presence of large amounts of CO2 indicates (to me, at least) that your plants aren't doing a very good job of absorbing it and converting it into oxygen.
Which leaves sunlight. And lots of it. Photosynthetic plant life will be even more abundant on your planet than it is on Earth.
However, you also note that you don't want everything to be green. No problem: there are many different pigments that Earth plants use for photosynthesis, with green chlorophyll only being the most common. See the table here: chlorophyll a is the familiar green stuff, chlorophyll b is yellow, chlorophyll c is blueish-green, chlorophyll d is found in red algae, and chlorophyll f absorbs far infrared light, so it probably appears colorless to our eyes.
No one's quite sure why chlorophyll a, which mostly reflects green light, is the most common chlorophyll on modern Earth. After all, the Sun produces more green light than any other color, so it seems like the most common types of plants would be adept at absorbing green light.
(source)
Perhaps the first photosynthetic bacteria to evolve on Earth were good at absorbing green light, and blanketed the surface of the oceans. Other bacteria evolved chlorophyll a to take advantage of the red and blue light the surface microbes left behind; and then the purple surface bacteria all died out... for some reason. Maybe. Or maybe chlorophyll a just popped out of a random mutation, worked well enough, and evolution ran with it. There's no way to know.
If you want polychromatic flora, just go for it. There's no reason why plants that evolve on another planet need to look just like the ones here.
