There's no need to count the number of enemies in each area. Just keep killing enemies in the area until you get the "But nobody came" message in your random encounters. This means you have killed all the enemies in this area and you can continue on to the boss. Be sure you finish this before approaching the area's boss, even if you can normally go back before actually fighting them.
Save points outside the Ruins also have a counter indicating how many enemies are left in the area.
When doing a Genocide run don't forget killing the dummy and the initial Froggit.
You can view a full list of all the enemies you need to kill here.
The Ruins save points don't give a kill counter, only later ones do.
You need to keep killing more monsters until battles give the message "But nobody came..." This must be done before approaching the boss monster for the area (at the point where you would normally fight them), even if you could normally turn back without fighting them. As long as you're on the Genocide route, save points checked outside the Ruins will give you a count of the number of monsters you have left to kill in the area. The count of monster kills for each section, followed by the name of the area boss, is as follows.
- Ruins: 20 kills (Toriel)
- Snowdin: 16 kills (Papyrus)
- Waterfall: 18 kills (Undyne the Undying)
- Hotland/Core: 40 kills (Mettaton NEO)
You don't need to kill every enemy you encounter, you're allowed to flee from or spare enemies, except for a select few), but you must also kill every unique enemy you encounter. This includes Shyren and Snowdrake. Although the Shyren encounter will happen even if the kill counter is depleted, the Snowdrake encounter will not and can ruin a genocide run.
Note that the requirements is in the number of kills, not battles, so battles with more enemies are more efficient for quickly reaching your goal. You will know that you have enough kills when interacting with a save point simply produces the message "Determination". If you ever leave the Genocide route, there's no way to get back on it without reloading a save from before you left the route.
Source: http://undertale.wikia.com/wiki/Genocide_Route#Requirements
Best Answer
No. Sparing monster kid will place you back on a neutral path, triggering the normal Undyne fight in the next room, instead of the Undyne the Undying fight there.