Sorry, I was thinking "spa" because of your leftover chemicals for the spa. Yes, Trichlor pucks/tabs are very acidic. For every 10 ppm Free Chlorine (FC) added by Trichlor, it also increases Cyanuric Acid (CYA) by 9 ppm and decreases Total Alkalinity (TA) by 7 ppm.
As for adding concentrated chemicals to the pool, it is best to add them slowly over a return flow in the deep end with the pump running. To ensure thorough mixing, lightly brush the side and bottom of the pool in the area where you added the chemical. Your circulation system should then distribute the chemical through the rest of the pool. There are some chemicals you do not want to put into the skimmer, such as Trichlor pucks/tabs (except for some special ones that don't dissolve when there is no water flow) so it's safest to add them over a return flow. For some chemicals like baking soda used to raise TA, it's fine to distribute it around the pool and most algaecides, enzymes, clarifiers, etc. say to distribute around the pool as well.
Wait in between your chemical additions and NEVER mix concentrated chemicals together. As for adding stabilizer, it dissolves slowly so you can put it in a sock and hang it over a return flow or you can add it slowly in the skimmer to get caught in the filter though the latter will take days to dissolve. I use an old T-shirt I put over my skimmer basket and add the CYA to it, but I also have floor drains and a skimmer bypass inlet so there is no risk of blocking the suction input to the pump.
For your pool using Trichlor, your TA level is way too low which is why your pH tends to drift too low. Though you can add baking soda, you can also add some pH Up (sodium carbonate; Arm & Hammer Super Washing Soda) to raise both pH and TA though you'll also need more baking soda since your TA is so low.
Since you have been using Trichlor pucks, I doubt that your CYA is really 0. However, you also show that FC is 0 so why is that? Did you just open the pool over the winter? If so, then if you closed with CYA being present then bacteria could have converted it to ammonia which can create a huge chlorine demand. I suggest you read the Pool School
for more info on opening your pool and maintaining it. If you use only Trichlor puck/tabs, then the CYA level will rise and if you don't proportionately raise the FC level then algae can grow faster than chlorine can kill it unless you use supplemental algaecide or phosphate remover at extra cost. The alternative is to get the CYA to the right level initially and then use chlorinating liquid or bleach as your primary source of chlorine. The downside is that this usually requires manual chlorine addition every day or two unless you've got a mostly opaque pool cover.
Also, I doubt that your hardness is really 0. Are you using test strips? They are next to worthless. You should get yourself your own good test kit -- either the Taylor K-2006
or the TFTestkits TF-100