Power surge, last paragraph.

“Once per turn when you deal damage to a creature or object with a wizard spell, you can spend one power surge to deal extra force damage to that target. The extra damage equals half your wizard level.”

when exactly is a wizard considered to be causing damage with a spell? Obvious answer seems to be when the wizard uses actions in some way to cause damage. But what about ongoing spell effects?

Wizard casts Fire bolt that hits a creature, seems to work.

wizard casts create bonfire, and a creature ends its turn in that area for some reason and fails its dexterity save on its turn. Is the wizard considered to have “dealt” that damage?

wizard casts wall of fire, and on a subsequent turn an ally pushes an enemy creature through the wall after a successful check. Is the wizard considered to have “dealt”  that damage?

wizard casts magic weapon on an allies no magical weapon. On the allies turn, they successfully hit an enemy dealing damage. Is the wizard concentrating on the spell contributing enough to be considered to have “dealt” the 1 bonus damage?

power surge is worded in a way that seems like the intent is to replicate empowered metamagic mixed with empowered evocation, but the trigger of the bonus damage seems vague with the way that spells don’t seem to define outright what causes the damage.

The generally accepted answer is very nuanced and only loosely based on actual RAW we actually have.  I'll go over your examples:

• Create Bonfire:  The wizard dealt the damage, but is the bonfire a spell?  Because the Bonfire is an object, the generally accepted answer is yes. 
  • Wall of Fire:  This is identical to Create Bonfire in all respects.
  • Due to the need to reconcile some otherwise self-contradictory RAW, I would answer no for Animate Dead, Animate Objects, or Summon Fey, because they are creatures.
• Magic Weapon:  Now you're getting into a more serious rules conundrum, but because of the wording on the RAW on Life Domain Clerics using Goodberry, the only consistent answer is also yes: just as a Life Domain Cleric's Goodberry counts as the Cleric using a spell to restore hit points, despite, in general, someone else eating the berry, your Magic Weapon must also count as you using a spell to deal damage, despite someone else wielding the weapon.

The rules that have to be reconciled are these:

• Life Domain Clerics' L1 Domain power must work on Goodberry, or your ruling is violating the RAW.  This means an object created by an instantaneous spell is still a spell later, despite the spell that was cast being over.  In your case, it also means you count as using the spell when someone else uses the object.
• Attacks made by a summoned creature are only magical attacks if their statblock says so, but attacks made by a spell are by definition magical - these are two instances of the RAW with equal weight.  Any ruling must be consistent with these as well, or it violates the RAW.  This means a summoned creature is not, itself, a spell, even though a summoned object is. 

After that, the RAW quickly starts fading into nonexistence.  Magic Weapon is almost exactly like Goodberry, but what about Haste or Polymorph?  In the absence of actual RAW - which we do not have - the generally accepted distinction to draw is object vs creature.  If you manifest an object or buff an object, that object (or more esoterically, the buff on it) is functionally your spell, temporarily.  If you manifest a creature or buff a creature, no such rule applies.  This isn't based on RAW so much as people in general trying to play the game with the rules we have and trying to be as consistent as possible. 

 when you deal damage to a creature or object with a wizard spell

it might just be arguing semantics but if you cast Create Bonfire and deal the damage to a target by having the bonfire appear under their feet or catch someone in your Wall of Fire area of effect then you are dealing the damage with a wizard spell.

if someone pushes a target into your spell's area of effect I would not count it as you dealing the damage.

Magic Weapon spell doesn't deal any damage.

not sure if you're of this quote or not but when asked what art was someone (famously?) replied... "I may not be able to describe what is art but I know it when I see it."

...or something to that effect... it works sort of the same here, I'm willing to bet just about every wizard spell can be determined that the wizard is dealing the damage with the wizard's spell or not.

some examples:

1. you cast Fire Bolt at a target who has climbed onto a roof... the surge can add extra damage to this (Yes)

2. the fiery roof collapses a few rounds later doing damage to those inside... the surge cannot add extra damage here (No)

3. you summon an elemental, any damage it deals is considered damage it dealt... the surge cannot add extra damage here (No)

Magic Weapon deals 1 point of damage in the same way Goodberry heals 1 hit point.

Funny thing is... it doesn't work that way for Magic Weapon

Magic Weapon only makes a mundane weapon magical. Even if you yourself used the weapon it would not count as a magic spell dealing damage. It would count as a magic weapon dealing damage and not be usable with the surge.

I'd let bonfire, wall of fire, flaming sphere, etc work. Even if the damage is done later, that damage is directly caused by the spell effect.

Spells that create or enhance creatures or objects with their own stats don't directly cause the damage, so don't count.

Seems pretty divisive. Would be cool to get some sort of clarification. I tweeted at Mr Crawford, so far no luck.