1) Flourish Excel is not a detergent. It's primary ingredient is gluteraldehyde, a chemical sterlizer used to sterilize heat sensitive medical and dental equipment. It's said to break down in the aquarium to produce a carbon source for plants. It kills algae and some plants are very sensitive to it. I believe that it can be very hard on livestock. I have used it in the past to fight Blackbeard algae, but I don't like it and I don't use it normally!
Any/all plants can/will grow without pressurized CO2, however, growth will typically be slower and overall plants sizes may be reduced some.
2) Low Tech/High Tech are terms used to define planted tanks. A low tech tank is like yours, mine, and Byron's. Low tech has somewhat lower light requirements, does not use pressurized CO2, and requires less fertilizer.
So the high tech tank uses brighter light (or a longer photo period), CO2, and more fertilizer - pushing plants to grow faster. With the possible exception of many carpet plants (that typically grow in shallow water, witch means bright light) many/most plants will grow in a low tech tank. Your baby tears may or may not be a bit of a challenge in low tech.
3) Where possible it's always a good idea to remove algae manually, however trimming may or may not be the answer as some plants react differently to trimming. You may be able to remove affected leaves, leaving the rest of the plant intact. Unfortunately, In some more extreme cases, entire plants must simply be removed to eliminate the algae from the system. In other cases, H2O2 (hydrogen peroxide) or Excel may be used as a spot treatment. For example, I have removed java ferns and anubis nana from the tank, sprayed with H2O2, rinsed and returned to the tank to kill BB algae. HOWEVER, not all plants can hold up with this treatment. (I tried this once on an Amazon Sword, and in the next few days, the leaves turned yellow). Although the plant may have recovered (as I did not spray the roots), I removed it for appearance sake.