First of all, welcome to the Pool Forum. Glad you found us!
To answer some of your questions:
1. Depends. Some pools need bleach every evening, others can go a day or two between doses. It really depends on several on several things --- your CYA level, the weather, how much of a swimmer load, how much organic debris drops in your pool, etc. I usually test mine every day although sometimes it might be every other day. I can usually add bleach every other day but sometimes every day. With your high CYA, you may be able to go several days between doses of bleach.
2. You only need to shock the pool if you have a measurable CC (combined chlorine) level. You won't be able to test this unless you buy a good kit. We recommend everyone buy a good kit. The one we suggest is the Taylor K-2006 or 2006C which you can get through the test kit link in my signature below.
Since the good kit is only available online, you need to get something to use in the meantime. See if your Walmart has an HTH 6-Way kit (not test strips) or if not, get a cheap OTO/Phenol Red kit (yellow and red drops).
3. Did your pool company use test strips or the "disappearing black dot test" to test your CYA? If you get a good kit, you'll be able to test your own CYA. You are right that the most likely reason you got algae is from not keeping the chlorine high enough. You need to follow the levels listed in the Best Guess Chlorine Chart which is also in my sig.
Either one of the two kits I suggested above that you should try and get until your ordered kit comes in will only test chlorine levels up to 5ppm. With such high CYA, you'll need much higher chlorine levels than that. You can force those kits to read higher by using a dilution method explained here: Testing Without a Good Kit
4. It is fine to add the bleach to the skimmer while the pump is running. That is the way I always add mine. If you have as high of a CYA reading as you report, then you should not be using those tabs. They are stabilized which means they have CYA in them. Let the ones that are wet finish dissolving and then no more. Also, no dichlor shock powder. Just use bleach for your source of chlorine.
Do a lot of reading here on the forum. (Until your registration is complete, you'll have to log out to be able to see the rest of the forum.) Also, go and read at our sister website www.poolsolutions.com. Lots of good stuff there, too.
Hope this help you. Let us know if you have further questions.
EDIT to add: Please put your pool's information in our pool chart. This makes it easier to help.
Pool Chart Entry Form
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