This item is overpowered
Let's go point by point:
Healing: This is roughly equivalent to casting Healing Word for free every round. Concentration is much cheaper than a bonus action+spell slot, and it ensures that your party members can literally never die if they're near you. Consider that the Paladin, who is focused on buffing auras, never gets a healing one. Additionally, low-level characters have very little health, so 2hp is actually quite a lot.
Swiftness: This is a weaker version of Longstrider. I have never seen anyone use that spell, and extra movement is generally not that useful (it doesn't affect CR, for example).
Damage: This, along with the pushing effect, is equivalent to a successful-save thunderwave. However, as the damage increases, it becomes almost strictly better than casting the spell.
The fundamental issue with this item is that it allows you to cast the equivalent of 1st level spells without requiring spell slots. A 1st level bard only has 2 1st level spell slots, but one with this item effectively gets an infinite number. This is particularly problematic with the healing, as this results in unlimited healing out of combat, making things like traps and attrition worthless.
Balancing the item
Consider the DMG's advice on magic items (285):
Maximum Spell Level. This column of the table indicates the highest-level spell effect the item should confer, in the form of a once-per-day or similarly limited property. For example, a common item might confer
the benefit of a 1st-level spell once per day (or just once, if it's consumable). A rare, very rare, or legendary item might allow its possessor to cast a lower-level spell more frequently.
Your item lets your player cast lower level spells infinitely.
The best way to fix this is to introduce a fixed number of charges. Many staves in the DMG have a fixed number of charges, and can expend them to cast various spells. For example, the staff of frost (DMG 202):
The staff has 10 charges. While holding it, you can u e an action to expend 1 or more of its charges to cast one of the following spells from it, using your spell save DC: cone of cold (5 charges), fog cloud (1 charge), ice storm (4 charges), or wall of ice (4 charges).
You could decide that your item starts with, say, 1 charge, and it costs 1 charge to use the song of swiftness, 2 charges for damage, and 3 charges for healing. As your character levels up, the item is able to have more charges, and can thus access the more powerful abilities.
Adding charges mostly solves the overpowered-ness of the item and adds a built-in method for progression, which can replace the more convoluted progression that you have now.
A few more small things: while it's implied, you should explicitly state that each use of the mandolin's powers costs an action to use, and give them a defined duration (note that lots of concentration spells still have finite durations). This will save you lots of headaches from creative players in the future. Finally, while it's not totally necessary, almost all sources of damage in the game are die rolls--you can replace 2 damage with 1d4, for example, to be consistent with the rest of the system.