Your GM is wrong
The general rule is that, when attacking with a melee weapon, the character can add their Strength modifier (or optionally Dexterity modifier, for Finesse weapons) to damage rolls. From Player's Handbook, page 196, "Damage Rolls":
When attacking with a weapon, you add your ability modifier -- the same modifier used for the attack roll -- to the damage.
From page 194, "Modifiers to the Roll":
The ability modifier used for a melee weapon attack is Strength, and the ability modifier used for a ranged weapon attack is Dexterity. Weapons that have the finesse or thrown property break this rule.
The spell conjures, explicitly, a weapon, but makes no specific exception to the general rule, so the normal logic for attack and damage rolls is used. Since the weapon has the Finesse property, you can and must use either Strength or Dexterity modifier to make the attack and damage roll.
It's also worth noting that all the weapons in the Handbook are described by their damage dice only; for example the greatsword is listed as dealing "2d6" damage, not "2d6 + strength modifier".
Yes
It counts as a simple melee weapon with which you are proficient.
+
If you're proficient with a simple or martial melee weapon, you can use it as a spellcasting focus for your bard spells.
As written, the Shadow Blade satisfies the requisites for being used as a spellcasting focus. The text from College of Swords does not require it to be an actual/physical weapon, simply a melee weapon with which you are proficient.
Best Answer
The Shadow Blade doesn't exist long enough.
The duration of shadow blade is 1 minute (concentration). Pact of the Blade (PHB. 108, emphasis mine) states:
Even assuming the ritual does not take concentration or you would have some way to circumvent it if it did, the shadow blade does not exist long enough to finish the ritual required to bind it.