The magic stone always deals the same damage regardless of launching method

The relevant parts of the spell text are:

You or someone else can make a ranged spell attack with one of the pebbles by throwing it or hurling it with a sling. [...] On a hit, the target takes bludgeoning damage equal to 1d6 + your spellcasting ability modifier.

The normal damage of the sling is irrelevant, because you are not using it to make a weapon attack. Instead, you are making a ranged spell attack, which deals damage as described in the spell.

In your example, the damage will be 1d6+3.

What if I'm using a +1 magic sling?

A +1 weapon states:

You have a +1 bonus to attack and damage rolls made with this magic weapon.

Since this text doesn't limit the bonus to only weapon attacks, this +1 bonus applies to both the ranged spell attack roll and damage roll of a magic stone launched using the sling.

What if the target is more than 30 feet away?

The rules for the range weapon property say:

Range. A weapon that can be used to make a ranged attack has a range in parentheses after the ammunition or thrown property. The range lists two numbers. The first is the weapon's normal range in feet, and the second indicates the weapon's long range. When attacking a target beyond normal range, you have disadvantage on the attack roll. You can't attack a target beyond the weapon's long range.

As with the +1 weapon case, the rule for weapon range does not limit itself to only weapon attacks, so the disadvantage beyond the sling's normal range also applies when using the sling to make a ranged spell attack with the magic stone.

The same principles can be applied to any other questions about which properties of the sling apply to the magic stone attack and which ones don't: if the property applies only to weapon attacks, it is not relevant, but if it applies to all attacks made with the sling, it applies to the magic stone attack.

(5 × Level) + your Int + defenders Con

Interpreting equations like this isn't defined by the game rules and so we use the english meaning. Unfortunately, english (and most other languages AFAIK) isn't very good at it, which is a big reason for using mathematical symbolics. As a good rule; take the simplest reading as you scan through. This gives:

$$ 5 \times \text{Level} + \text{Your Int} + \text{Defender's Con} $$

which for anyone not fully comfortable with order of operations equals

$$ (5 \times \text{Level}) + \text{Your Int} + \text{Defender's Con} $$

If it were to mean

$$ 5 \times ( \text{Level} + \text{Your Int} + \text{Defender's Con} )$$

it should (in my opinion) been worded as (italics to emphasise change):

equal to five times the sum of your level in this class + your Intelligence modifier + the Iron Defender’s Constitution modifier

As additional support, the same problem of having such equations in D&D rules have appeared for Arcane Ward where lead designer Jeremy Crawford has chimed in on twitter (not official ruling, but gives what the intended meaning is) with the statement that

The ward has hit points equal to twice your wizard level + your Intelligence modifier.

should be read as:

its hit point maximum equals your Intelligence modifier plus twice your wizard level.

Meaning to be read as

$$ (2 \times \text{Level}) + \text{Int} $$

similar to that above.

As an addendum: the final version of the subclass clarifies the calculation as:

equal the steel defender’s Constitution modifier + your Intelligence modifier + five times your level in this class

and this is how the feature is implemented in D&D Beyond. (Thanks to @Garret Rooney and @V2Blast)