Level 50+

If your dinoz is level 50 you can unlock level 60 by speaking with Sopie.
See Caushemesh quest and Cassandre missions.

The Villa
Sofia
Villa (Dino Plaza)

I can see that in your current state we're not going to get very far, and it would appear that it'll be many moons before we can do anything about that... OK, I can free your dinoz from its current limit, but be certain this is what you wish to do, as this decision is permanent! Once I perform the limit break, you will be able to evolve all the way to level 60!

After you unlock level 50+ skills your dino receives 1st Limit Break status - Your dinoz has realised its potential, it can now go beyond level 50 and level up as far as level 60!

However, when they are exposed to the ether for a short time, they can learn new skills which make them much stronger! Be aware though, after this is done, the dinoz will no longer be able to access the old skills trees so you will not be able to go back and learn any missing skills further down the line... before you expose your dinoz to the ether, make sure you have optimized your dinoz skills...

If you choose new skills path, your dino receives Drop of Ether status - Your dinoz bathed in pure ether, it can now evolve using a whole new skills tree!

When you finish Celestial Temple quest, you get access to Rhubabao'riley After all the missions, talk to Rhubabao'riley to get lvl 70 unlock status and unlock level 70.

Arrival of level 80! This will be the last level rise for your Dinoz.
At level 69 there are available Master Zenith's missions. When you finish all the missions, you get lvl 70 unlock status and unlock level 70.

This list of skills is taken from a thread 'Lvl 50+ New Skills', which you can find on the official forum.
Here is spreadsheet.
You have to pay attention, some skills have been translated and may not be the exact words used, text descriptions aren't official ones, some may be part observations and not an actual translation of the descriptions.

Air

Level 1 Level 2 Level 3 Level 4 Level 5
Mind over Body All White on the Night Double Face Uvavu Qi Gongb
Whips Twinoid 500mg Sylphs
Anaerobics Angel Breath Harvester Messiah
Hurricane Hades Store Mutiny
Heat Sink Sticking Hands

1 Mind over Body P Increases a dinoz healing ability by 25% -
2 All White on the Night P +20 assault for the dinoz while being in Nimbao -
2 Anaerobics P -25% endurance -
3 Double Face P At the beginning of the fight +/-10 initiative (50%) -
3 Whips P -15% healing ability -
3 Angel Breath P +20 fire defense -
3 Hurricane P +20 air assault -
4 Uvavu A All Dinoz with an Air element of less than 10 points will lose 50% of their speed for the next three turns high
4 Twinoid 500mg P +50% endurance -
4 Harvester C +1 gathering square -
4 Hades Sore S +1 equip square -
4 Heat Sink A Stop fire (spheral) A/E sills
5 Qi Gong E Takes away all the endurance of an enemy very high
5 Sylphs A Removes one opponent dinoz from battle
5 Messiah U +3 active dinoz on your account -
5 Mutiny E Causes clones to attack their masters high
5 Sticking Hands E Blocks an enemy's ability to dodge attacks weak

Fire

Level 1 Level 2 Level 3 Level 4 Level 5
Dinoz Max-Protein Attenuator Magma Shell Burning Fever Ember Country
Artemesian Prayer Rocky Receptacle
Phoenix Feather
Deep Red Battle Cry Flaming Armour Brazier Fist
Joker Frat Party

1 Dinoz Max-Protein P +30% endurance -
2 Attenuator A Reduces the enemy's ability to recover by 25% weak
2 Deep Red P +20 assault for the dinoz while being in the Big-All-Hot -
3 Magma Shell P +30HP and +5 fire assault -
3 Battle Cry E Next turn your dinoz will launch his best A-skill high
4 Burning Fever P +30% multi-hits -
4 Artemesian Prayer C Additional hunting square -
4 Flaming Armour P +20 wood defense -
4 Joker P +25% or -25% speed at the beginning of the fight -
5 Ember Country E All Dinoz with a Fire element of less than 10 points will no longer use elements A or E for the next three turns.
5 Rocky Receptacle A Negates A/E Wood sphere skills on one dinoz
5 Phoenix Feather E? Survival (allow dinoz to escape one death in combat), then gains +10 initiative/speed, but dinoz cannot heal back
5 Brazier Fist P +20 fire assault -
5 Frat Party E +30% healing ability on all your dinoz

Lightning

Level 1 Level 2 Level 3 Level 4 Level 5
Moral Support Yellow Fever Chronic Cramp Electric Fence Force of Zeus
Oracle Love Hertz
Cardiac Massage Sun Blindness Backup Battery Air Receptacle
Einstein St Elma's Fire

1 Moral Support P +10% endurance -
2 Yellow Fever P +20 assault for the dinoz while being in the Steppes -
2 Cardiac Massage P Increases a dinoz healing ability by 20% -
3 Chronic Cramp E -10 energy points, and reduces a dinoz healing by 15% weak
3 Sun Blindness E Gives "Blind" status to opponent dinoz (miss 1 assault on 3) normal
4 Electric Fence P +20 air defense -
4 Oracle P Doubles Invocation launch rate -
4 Backup Battery P +50HP but -15 initiative -
4 Einstein C Dinoz can energize once more -
5 Force of Zeus P +20 lightning assault
5 Love Hertz P +20% dodge bonus -
5 Air Receptacle A Negates A/E air sphere skills on one dinoz
5 St Elma's Fire A All Dinoz with a Lightning element of less than 10 points will lose 5% of their HP for the next three turns.

Water

Level 1 Level 2 Level 3 Level 4 Level 5
Holy Water Deepest Blues Chromatic Diet Abyss Tesla Receptacle
God Banned Marine Vitality
Gamma Radiation Aqueous Metamorphosis Aphrodisia Magic Tourbillon
Nemo Hyperventilation
Armor-Plated Shell Cleptomania Group Therapy

1 Holy Water S Doubles healing HP while resting -
2 Deepest Blue P +20 assault for the dinoz while being in Atlantean Islands -
2 Gamma Radiation P +5 on all assaults, +30HP and -10 initiative -
3 Chromatic Diet E An opponent dinoz get "monoelement" status (dinoz can only attack with one element) high
3 Aqueous Metamorphosis P +20 lighting defense -
3 Armor-Plated Shell P +10 armor, -20% speed -
4 Abyss A All Dinoz with a Water element of less than 10 points will see the strengh of their attacks and assaults drop by 25% for the next three turns high
4 God Banned E Opponent dinoz get "mute" status
4 Aphrodisia S Another dinoz can follow -
4 Nemo C Can fish an additional square -
4 Cleptomania P Forbids one opponent dinoz to use non-magic items once -
5 Tesla Receptacle A Negates A/E Lightning sphere skills on one dinoz high
5 Marine Vitality P +80HP -
5 Magic Tourbillon P +20 water assaults -
5 Hypervenilation A Decrease the endurance of all enemies by 20%
5 Group Therapy E Holder heals back as much as the opponent team is healed from a skill

Wood

Level 1 Level 2 Level 3 Level 4 Level 5
Muscular Oxygenation Life Source Rock Thrower Re-Searcher Mud Wall
Steel Skin
Energy Sink Aches Flow of Life Skill Forger
Racing Green Shield Dinoz Force Control Berserk Amazonia
Lactic Acid Flame Skin River of Life Aqueous Receptacle

1 Muscular Oxygenation P +20% Endurance -
2 Life Source S If the dinoz is hit by an assault, it can absorb up to 5% of the life points of their enemy weak
2 Energy Sink S Reduces an adversary healing capacity by 15% during an assault weak
3 Rock Thrower A Shoots a rock at one dinoz with wood*10 high
3 Aches E -30% opponent dinoz endurance on an assault weak
4 Re-Searcher C Additional excavating square -
3 Flow of Life P +20 water defense, +20 wood assault -
5 Mud Wall A Absorbs up to 30 dmg points before vanishing. Vulnerable to the KSB.
5 Steel Skin P +100HP -
5 Skill Forger E Copy one used skill by opponent and keep it until the fight ends
1 Racing Green P +20 assault for the dinoz while being in the Grumhel Forest -
2 Shield Dinoz E Protect a dinoz with the less HP by taking their damage in the whole fight
2 Lactic Acid P -15% Endurance -
3 Force Control E? Dinoz deals at least 10dmg on an opponent with assault
3 Flame Skin P +50HP -
4 Berserk E Doubles assault power, but forbids A and E skills use
4 River of Life P +20 initiative -
4 Aqueous Receptacle A Negates A/E Water sphere skills on one dinoz
5 Amazonia A All Dinoz with a Wood element of less than 10 points will sleep for the next three turns unless they are subjected to an attack which causes them to lose 10HP