Assuming like TA is like DA, espionage comes in two flavors: passive and active.  You can probably get a better and fuller description at the galciv2 wiki (Google is your friend) but here goes.

If you go to the espionage screen, you will see that each race has a box with 4 slots or openings.  If you have an unassigned spy, toggle the spot under the box that places the spy.  To remove or reassign, toggle that box.  This is passive spying, and those spies cannot be neutralized/killed.

As you build up spy weeks, you will be notified of the level on each race.  Eventually, you can "see" the destinations of most enemy ships and fleets of that AI.  This is useful on the defense.  Note, though, that once you get to the higher levels, you only gain additional "spy weeks" while 3 or even all 4 boxes are occupied.  Once you get to "Advanced" you can remove the spies, as there will be no further espionage gains.  Other info includes knowing all the enemy techs, what they are researching, etc.

Be aware that if a race ceases to exist while you have spies in that race "box", then they are lost to you forever, having become "archeologists".  This is a known bug.  Search here on that word to get the joke.

You seem to know of the "Active" spy uses, and that they can be neutralized.  While an active spy is in place, you negate the structure it is "on" while still requiring its maintenance costs be paid.  You also have some random chance each turn of stealing a tech that that race has, and which improvement or which planet on which you have placed that spy does not affect that, simply that it is there.

If I got it wrong, I'm sure someone here will straghten us both out.