Do Mini-Sentry Guns count for the Gun Down achievement

You will not get the achievement if the enemy sentry is in the construction process and it must be a regular sentry. Your sentry on the other hand can be any kind.

The Wrangler allows your Sentry Gun to shoot at an enemy Sentry Gun from outside of its range, unless the enemy Engineer is using the Wrangler as well. On the offense, try to haul your Sentry Gun into a position where you can easily attack the enemy Engineer's, the element of surprise is often enough to force other players to abandon their building nests. Unless your Sentry Gun is level 3, try to make sure the enemy Engineer is not nearby, or kill him or her before attempting to destroy the Sentry Gun, as the Engineer will be able to repair any damage done to it otherwise. Be on the lookout for other Engineers trying to deploy their Sentry Guns around corners and use the rockets of a level 3 to destroy them. Destroying a Combat Mini-Sentry does not count towards this achievement.

http://wiki.teamfortress.com/wiki/Obtaining_Engineer_achievements

Only melee weapons. So, any usage of any rifle or pistol (especially when you get incapacitated), including the grenade launcher will deny you of the achievement, even if you don't do any damage.

Chainsaw is a melee weapon, too, by the way.

Throwing molotov, pipe, bile jar or shoving zombies around (even killing them with a gun only by shoving) will have no negative effect. Also interesting is that usage of machine guns and stationary turrets won't cancel the achievement either, since they don't count as guns, but part of the environment.

HOWEVER, if you fire one bullet, you have to restart the campaign ALL over. Dying and repeating the map will not reset this (have actually tried it).