Not many issues here.

1. Culling Blade's description says that the aoe boost applies when killing a hero. But it also applies when killing non-hero units. The cooldown part works on heroes only.
2. Battle Hunger's debuff does not say that it slows. Only the damage is mentioned.
3. Culling Blade's buff does not say how much speed it grants, neither does Battle Hunger's buff.
4. Battle Hunger's description says that Axe gains ms for every unit affected by Battle Hunger. It should be more precise, since only units cursed by Axe himself grant ms. Rubick can curse units too and that doesn't make Axe faster


Besides these 4, there are not really any issues.

The ALT notes could be improved though.

1. Berserker's Call has a redundant note about spell immunity. This info is already in the header.
2. Battle Hunger has a redundant note about denying cursed units. Heroes can never denied (except when affected by 2 specific spells), and Battle Hunger does not prevent denying cursed units
3. All of his spells could make use of new ALT notes. Call and Culling Blade have none. You can see suggested notes in the image below.

Spoiler: 



Here are the modified sections of dota_english.txt required for these changes and additions. Changed values are marked red and new values are marked blue.

Code:
		//AXE
		"DOTA_Tooltip_ability_axe_berserkers_call"										"Berserker's Call"
		"DOTA_Tooltip_ability_axe_berserkers_call_Description"							"Axe taunts nearby enemy units, forcing them to attack him. Preparing himself for the incoming attacks, Axe also bolsters up his physical defenses."
		"DOTA_Tooltip_ability_axe_berserkers_call_Note0"								"If taunted units cannot attack, or Axe cannot be attacked, they follow him, staying within attack range."
		"DOTA_Tooltip_ability_axe_berserkers_call_Note1"								"The taunt is not canceled upon losing vision over Axe."
		"DOTA_Tooltip_ability_axe_berserkers_call_Note2"								"Cannot taunt buildings, wards and invulnerable units."
		"DOTA_Tooltip_ability_axe_berserkers_call_Note3"								"When Axe dies, the taunt ends immediately."
		"DOTA_Tooltip_ability_axe_berserkers_call_Note4"								"Axe gains the armor even when missing completely."
		"DOTA_Tooltip_ability_axe_berserkers_call_Lore"									"Mogul Khan's warcry taunts opponents into engaging in an unconquerable battle with the Axe." 
		"DOTA_Tooltip_ability_axe_berserkers_call_radius"								"RADIUS:"
		"DOTA_Tooltip_ability_axe_berserkers_call_bonus_armor"							"BONUS ARMOR:"
		"DOTA_Tooltip_ability_axe_berserkers_call_duration"								"DURATION:"
		"DOTA_Tooltip_ability_axe_battle_hunger"										"Battle Hunger"
		"DOTA_Tooltip_ability_axe_battle_hunger_Description"							"Plunges an enemy into a hunger for battle, slowing and dealing damage over time to it, until it kills another unit or the curse expires. Axe gains movement speed for each unit cursed by him until freed from the curse."
		"DOTA_Tooltip_ability_axe_battle_hunger_Note0"									"Destroying buildings or wards and denying allied units also removes the debuff."
		"DOTA_Tooltip_ability_axe_battle_hunger_Note1"									"Killing illusions does not remove the debuff."
		"DOTA_Tooltip_ability_axe_battle_hunger_Note2"									"Multiple casts on the same target do not stack."
		"DOTA_Tooltip_ability_axe_battle_hunger_Note3"									"Kills done with wards do not remove the curse for their owner."
		"DOTA_Tooltip_ability_axe_battle_hunger_Lore"									"Ordinary heroes cannot withstand Mogul Khan's rage for battle, such that it injures them until it is satisfied."
		"DOTA_Tooltip_ability_axe_battle_hunger_duration"								"DURATION:"
		"DOTA_Tooltip_ability_axe_battle_hunger_slow"									"%MOVEMENT SLOW:"
		"DOTA_Tooltip_ability_axe_battle_hunger_speed_bonus"							"%SPEED BONUS:"
		"DOTA_Tooltip_ability_axe_battle_hunger_damage_per_second"						"DAMAGE PER SECOND:"
		"DOTA_Tooltip_ability_axe_counter_helix"										"Counter Helix"
		"DOTA_Tooltip_ability_axe_counter_helix_Description"							"When attacked, Axe performs a helix counter attack, damaging all nearby enemies."
		"DOTA_Tooltip_ability_axe_counter_helix_Note0"									"Missed attacks cannot proc Counter Helix."
		"DOTA_Tooltip_ability_axe_counter_helix_Note1"									"Counter Helix fully works while hexed."
		"DOTA_Tooltip_ability_axe_counter_helix_Note2"									"Does not proc on attacks from buildings and wards."
		"DOTA_Tooltip_ability_axe_counter_helix_Lore"									"Axe is the only reinforcement this army needs."
		"DOTA_Tooltip_ability_axe_counter_helix_radius"									"RADIUS:"
		"DOTA_Tooltip_ability_axe_counter_helix_trigger_chance"							"%PROC CHANCE:"
		"DOTA_Tooltip_ability_axe_culling_blade"										"Culling Blade"
		"DOTA_Tooltip_ability_axe_culling_blade_Description"							"Axe spots a weakness and strikes, instantly killing an enemy unit with low health, or dealing moderate damage otherwise. When killing an enemy with Culling Blade, Axe releases a war cry, increasing the movement and attack speed of nearby allies. Culling Blade is immediately ready to strike again when killing an enemy hero with it.\n\nUpgradable by Aghanim's Scepter."
		"DOTA_Tooltip_ability_axe_culling_blade_Note0"									"Culling Blade does not go on coldown upon killing a hero or an illusion."
		"DOTA_Tooltip_ability_axe_culling_blade_Note1"									"The area speed bonus applies upon killing any unit, not only heroes."
		"DOTA_Tooltip_ability_axe_culling_blade_Note2"									"Can kill heroes who are affected by spells which would prevent death otherwise."
		"DOTA_Tooltip_ability_axe_culling_blade_Lore"									"Mogul Khan is the embodiment of battle and fury, launching into a gruesome fatality against those who dare engage the Axe in combat."
		"DOTA_Tooltip_ability_axe_culling_blade_kill_threshold"							"KILL THRESHOLD:"
		"DOTA_Tooltip_ability_axe_culling_blade_kill_threshold_scepter"					"SCEPTER KILL THRESHOLD:"
		"DOTA_Tooltip_ability_axe_culling_blade_damage"									"DAMAGE:"
		"DOTA_Tooltip_ability_axe_culling_blade_speed_bonus"							"%KILL MOVE SPEED BONUS:"
		"DOTA_Tooltip_ability_axe_culling_blade_atk_speed_bonus_tooltip"			"KILL ATTACK SPEED BONUS:"
		"DOTA_Tooltip_ability_axe_culling_blade_speed_aoe"								"KILL SPEED RADIUS:"
		"DOTA_Tooltip_ability_axe_culling_blade_speed_duration"							"KILL SPEED DURATION:"
		"DOTA_Tooltip_ability_axe_culling_blade_speed_duration_scepter"					"SCEPTER SPEED DURATION:"
		"DOTA_Tooltip_ability_axe_culling_blade_cooldown_scepter"						"SCEPTER COOLDOWN:"
		"DOTA_Tooltip_ability_axe_culling_blade_aghanim_description"					"Decreases cooldown, increases kill threshold, and increases bonus movement speed duration."
		"npc_dota_hero_axe_bio"															"As a grunt in the Army of Red Mist, Mogul Khan set his sights on the rank of Red Mist General. In battle after battle he proved his worth through gory deed. His rise through the ranks was helped by the fact that he never hesitated to decapitate a superior. Through the seven year Campaign of the Thousand Tarns, he distinguished himself in glorious carnage, his star of fame shining ever brighter, while the number of comrades in arms steadily dwindled. On the night of ultimate victory, Axe declared himself the new Red Mist General, and took on the ultimate title of 'Axe.' But his troops now numbered zero. Of course, many had died in battle, but a significant number had also fallen to Axe's blade. Needless to say, most soldiers now shun his leadership. But this matters not a whit to Axe, who knows that a one-man army is by far the best."

		"DOTA_Tooltip_modifier_axe_berserkers_call"					"Berserker's Call"
		"DOTA_Tooltip_modifier_axe_berserkers_call_Description"		"Taunted.\n\nForced to attack a specific target."
		"DOTA_Tooltip_modifier_axe_berserkers_call_armor"			"Berserker's Call"
		"DOTA_Tooltip_modifier_axe_berserkers_call_armor_Description"	"Gaining %dMODIFIER_PROPERTY_PHYSICAL_ARMOR_BONUS% bonus armor."
		"DOTA_Tooltip_modifier_axe_battle_hunger"					"Battle Hunger"
		"DOTA_Tooltip_modifier_axe_battle_hunger_Description"		"Afflicted with a terrible killing hunger. Slowed by %dMODIFIER_PROPERTY_MOVESPEED_BONUS_PERCENTAGE%%% and taking damage over time until killing a unit."
		"DOTA_Tooltip_modifier_axe_battle_hunger_self"					"Battle Hunger"
		"DOTA_Tooltip_modifier_axe_battle_hunger_self_Description"		"Gaining %dMODIFIER_PROPERTY_MOVESPEED_BONUS_PERCENTAGE%%% bonus movement from afflicted units."
		"DOTA_Tooltip_modifier_axe_counter_helix"					"Counter Helix"
		"DOTA_Tooltip_modifier_axe_counter_helix_Description"		"20%% chance to counter incoming attacks."
		"DOTA_Tooltip_modifier_axe_culling_blade_boost"					"Culling Blade"
		"DOTA_Tooltip_modifier_axe_culling_blade_boost_Description"		"Inspired by a war cry. Gaining %dMODIFIER_PROPERTY_MOVESPEED_BONUS_PERCENTAGE%%% movement speed and %dMODIFIER_PROPERTY_ATTACKSPEED_BONUS_CONSTANT% attack speed."