Final Fantasy 7 – Do Summon Kills Count Towards Limit Breaks?

Andrew Natoli had the fundamental concept down (more hits = more damage), but here's the actual list, in order of maximum potential damage:

1. 18 hits: Barrett's Angermax/Ungarmax and Cid's Highwind
2. 15 hits: Cloud's Omnislash and Yuffie's Doom of the Living
3. 13 hits: Cait Sith's Slots; Bar/Bar/Bar has a very small chance to summon Knights of the Round
4. 10 hits: Barrett's Catastrophe, Red XIII's Starburst Ray, and Yuffie's Bloodfest
5. 8 hits: Cid's Big Brawl
6. 7 hits: Tifa's Final Heaven (assuming you're lucky)
7. 6 hits: Cloud's Meteorain and Cid's Dragon Dive
8. 5 hits: Red XIII's Earth Rave and Vincent's Hell Masker (Splattercombo)
9. 3 hits: Cloud's Cross-slash

Notes:

• Since Tifa's limit breaks build on each other, she can hit for 1-7 times.
• All other limit breaks only hit once (or not at all, in the case of Aeris's Limit Breaks).
• Most of the multi-hit limit breaks hit multiple targets, so for maximum damage against one target, only one target should be alive.
• Limit breaks, especially the multi-hit ones, have a modifier so each hit does less than the maximum potential damage. But maximum potential damage isn't affected by the damage cap (that is, the damage an attack does gets rounded down to 9,999 after all modifiers are applied), so a fully-powered character can have each hit in the limit break hit for 9,999. Otherwise, consult this wiki page for information about the damage multipliers that'll affect each limit break.

Yes and No you can unlock level 4 without fully unlocking all your limit breaks. For most characters (with the exception of Vincent and Cait Sith) there are 7 limit breaks. The first level in levels 1 to 3 are unlocked by how many enemies you have killed with that character.

For example take a look at Clouds table to gain limits from GameFaqs

Cloud Strife

Limit 1/1: Braver         | Used 8 times to learn Limit 1/2: CrossSlash

Limit 1/2: CrossSlash    | 120 kills made to learn Limit 2/1: Blade Beam

Limit 2/1: Blade Beam     | Used 7 times to learn Limit 2/2: Climhazzard

Limit 2/2: Climhazzard    | 200 kills made to learn Limit 3/1: Meteorain

Limit 3/1: Meteorain      | Used 6 times to learn Limit 3/2: Finishing Touch

Limit 3/2: Finishing Touch| Omnislash manual used to teach Limit 4/1: Omnislash

Limit 4/1: Omnislash      | All limits learned.

If you have 120 kills you will learn Limit 2/1, without ever using your limit break with cloud you will unlock level 1/2. If once again you make 200 kills with Cloud with out ever using 2/1 you will unlock 3/1. Now to learn Level 4/1 you would need to get 3/2, but it is possible to never learn 1/2 or 2/2 limit breaks.