I fully concur with 1077 on the livebearers, they all prefer slightly alkaline (I guess it is now termed "basic"?) water and will not fare well in acidic water, at least that has been my experience.
Given 1077's news on the Brita, I wouldn't use water that goes through one of these. As Aunt Kymmie has confirmed, they do seem to do something to chemically alter the water, as I also agree with her that carbon filtration would not lower the pH from 8 to 6. That's a change of 100 times the acidity/alkalinity of the water.
Re the plants. Yes, some types prefer slightly acidic to neutral pH water, but there are some that would thrive in your water. Vallisneria will grow like weeds, at least it did for me several years ago in a livebearer tank with dolomite to keep the pH in the high 7's and with one 15w fluorescent tube. That was a 15g tank too. You have a fair amount of light (which is why the algae is rampant, they can convert carbon from carbonates better than plants and in such bright light will readily do so) so you might want to reduce the lighting period. Once you get some plants growing, experiment a bit and find the balance. When light and nutrients are in balance, the plants will thrive and algae will not gain a foothold.
CO2 is one way to reduce the pH but myself I wouldn't go down that road until some other things have been tried. The rocks may have been part of the problem if they are calcium-based, so removing them may see the pH fall to the tap water level of 8 or a bit below. Once that occurs, I think you're all set for livebearers, and try some Vallisneria and other plants that are not so fussy about water pH. Some of the swords will do quite well as long as the water is not too alkaline, and with your bright light the stem (bunch) plants should be OK. There are several sites on the web with plant info that you can Google.