Minecraft – When I mined the hopper, it didn’t drop as an item; was this because I didn’t use a tool

You can enchant books instead, which always puts only a single enchantment on it, and then use the book in the anvil to apply it to a tool or weapon. You do have to go through a lot of books to get exactly the enchantments you're looking for, and it costs extra to transfer the enchant from the book to the tool on the anvil, but what you get for the loss of efficiency is precision in having exactly the enchants you want.

Note that you can also combine enchanted books of the exact same level on the anvil to get one book that has that enchantment level +1. So, you can take two Efficienty III books, combine them, and get one Efficiency IV book as the result. In this way you can slowly gather together the pieces for your ideal set of magic tools and, with enough XP lying around to go through this XP-inefficient process, get a set of tools to your precise specs.

This is the intended way of operation. As both hoppers try to push their first item each way, only one ends up moving.

To make hoppers pass the whole load before starting to pass it back, you need to block the next (opposing) hopper until your current one is empty. The standard way (that you mention yourself) is measuring the contents of the current hopper with a comparator, and feeding this signal into the next hopper.

Here's SethBling's hopper clock from Minecraftwiki:

Hoppers are directed clockwise, right now the items are moving from top left hopper to top right one, which is still locked by a signal from top comparator, and is already sending a lock signal to bottom right hopper.

For more designs, see this section on the wiki.