I've tested my soil and it is very low in N and P. What is the best way to solve this problem? In a 10' by 20' garden, I added 40 lbs of 16-10-16 last year and it tested the same this year.
You may want to try some rotted manure. It is one of the best ways to add nitrogen as it also adds organic material (good for water retention) and trace micro-nutrients as well. Potash is typically used to replace potassium.
Chemical fertilizers have a tendency to leech out faster than other sources. This may be why the soil did not test any better this year.
Will spreading 13-13-13 now in a just cleared garden be beneficial for next year's crops or not? And will it be fine to till it in now?
Yes, you can work this in now. It actually helps with mellowing the fertilizer in the soil, which means that it is less likely to burn plant roots when you plant there next year.
Is there any benefit in the garden if I empty my used tea bags there?
Mixed into the soil or added to compost piles, tea can actually help add both texture and acidity to the soil. So the answer to your question is yes.
When should I apply fertilizer to my (mostly native) perennials and shrubs? In the fall?
I would hold off until spring for fertilizing your plants. Right now they are getting ready for winter dormancy and fertilizing will only promote new growth, which would provie detrimental for the plants.
I have a medium size vegetable garden. I rough dig the whole area at the end of the growing season. Do you recommend using a fertilizer now with the hope that it will 'wash' in during the winter months, and if so, what should I use?
I would recommend adding a fertilizer now so that it can mellow into the soil. I personally would use compost or manure, but you can also use a balanced fertilizer like 10-10-10 or 20-20-20.
What is blood meal and how is it used in the garden?
This article will help you: https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/blood-meal-fertilizer.htm
I am just wondering if you could possibly give me some of the generic names of some of the things that you put on the garden? For example, what you could use for aluminum sulfate and some of the other chemicals that are used.
Aluminum sulphate is often used to adjust pH levels in soil. Of course, you can also use other this to adjust soil pH as these articles will explain: https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/raise-acid-level-soil.htm
https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/lower-acid-soil.htm
As for a list of generic names, this would be far too long. There are many things you can put in the garden for a myriad of reasons. Perhaps, if you narrowed this down to something more specific (i.e. fertilizers, pesticides, etc.), we could be of more help to you.