The following is largely opinion based since there is very little more than anecdotes to go by in our hobby.
The "tabs" are in pill-like gel caps. Once submerged beneath/near a plant, do these caps dissolve? Or will I have to dig them out?
They will dissolve. They are gel capsules, AKA, pills. They are usually 100% food grade, safe for human consumption.
Are these 100% safe for all fish, invertebrates (mainly my dwarf frog) and plants?
The capsules are definitely safe. The O+ inside is up for debate. It is best to keep it submerged under the soil. I don't believe it will harm them if it surfaces, but I can't imagine it being good for the fish to eat. With that said, mine have never more than nipped at a pellet that has risen above the soil and they steer clear of it. As far as plants, it should be safe. Most plants can be grown outside of water too, so I believe the ferts are pretty interchangeable.
How long does a tab last? How often would I be replacing them?
This is largely dependent on your plant base. For slower growing plants, they will last longer. Anywhere from 30-90 days is average from my experience. I replace mine every 3-4 months.
If I purchase 400, how long will that last me? Just trying to figure out how many I would need to purchase to have a month/6 month/year supply.
If you place them in a 6" grid, they will last you a lifetime. That is what I usually suggest. One every 6" in every direction. This comes out to 16 for a standard 55g footprint. Then replace them once every 3 months. So 64 per year. You'd have nearly 6 years of root tabs.
How long do they remain active while in storage? Do I need to purchase in small amounts and use within a certain time frame? Or can they sit in my storage box indefinitely?
I would look at a box of O+ for an expiration date, but I don't think they go bad. Just don't get them wet, they melt together.
Would I need to make any changes to my current dosing? Would using osmocote tabs replace my current routine? Or would the tabs compliment my current routine?
The root tabs will compliment any dosing. The liquid carbon is completely separate and the trace elements are always good. Root tabs are just another layer, often the least efficient since many plants feed from column dosing much easier. It depends on your plants and you'll be able to see the difference in their growth, color, etc in a few weeks. If you notice deficiencies after several weeks, it is time to replace the tabs.