So this happened during a draft tonight...

I attack my opponent with a Fire Urchin. After blocks, I cast the Invert half of Invert // Invent on it, let it resolve, and then Sure Strike. What is the urchin's P/T when this is all done? My opponent and I both came to the conclusion that it was an 8/1, but I was discussing this later on with someone else who claimed it should be 7/2. Which is correct and why?

Sure, I'll take a stab at it. I think I can answer this one correctly.
Fire Urchin is a 1/3

1) Casting Invert causes Fire Urchin's ability to trigger, putting it's +1/+0 boost on the stack above Invert. Result of ability resolution is a 2/3 Fire Urchin.
2) Invert Resolves, resulting in a 3/2 Fire Urchin.
3) Casting Sure Strike causes Fire Urchin's ability to trigger, putting it's +1/+0 boost on the stack above Sure Strike, resulting in a 4/2 Fire Urchin.
4) Sure Strike Resolves, resulting in a 7/2 Fire Urchin.

This all happens because Fire Urchin's trigger is on cast, not on spell resolution (such as an ETB effect). So the power boost will always resolve before the spell you cast. Subsequent instants played before resolution of spells and abilities would look like the following:

(Top of stack)
Fire Urchin's triggered ability
Instant spell
Fire Urchin's triggered ability
Instant or Sorcery spell
(Bottom of Stack)

I hope this helped.

Layers


The only layer that applies here is Layer 7, which further breaks down to:

Here we're looking at layer 7c, and 7e. The important thing is that things in layers are always applied in the order listed above.

So you have a 1/3 Fire Urchin.
You cast Invert. Fire Urchin's ability triggers, and gives it +1/+0 (2/3). This is layer 7c
Invert resolves. Power and toughness are switched. (3/2) This is layer 7e.
Cast Sure Strike. Fire Urchin's ability triggers again, and gives it +1/+0. Now this is where it gets a little weird; what the game sees here is a 1/3, with +2/+0, switched, in that order so it's a 3/3, switched (still a 3/3)
Sure Strike resolves, giving Urchin +3/+0. This also happens in layer 7c. So you have a 1/3, with +5/+0, switched. You have a 3/6.

The answer is, Fire Urchin is a 3/6. Invert is bad for Fire Urchin.

Viperion's analysis checks out.

It may help to think of it that continuous effects aren't "baked in" to the card for later modification; instead the effects remain floating there ever-present, and at any given moment we can find out the object's state by applying all effects through the layer system to the card text.

Another way to look at the effects from what Viperion laid out is that by the time all of these spells and triggered abilities resolve, we just have the following continuous effects to apply:

Card text: 1/3 with trample
Effects:

• Layer 6:
  • Add first strike (from Sure Strike)
• Layer 7c:
  • +1/+0 (from Fire Urchin)
  • +1/+0 (from Fire Urchin)
  • +3/+0 (from Sure Strike)
• Layer 7e:
  • Swap power/toughness (from Invert)

Applying those,

• After layer 6 we get a 1/3 with trample and first strike
• After 7c we get a 6/3 with trample and first strike
• After 7e we get a 3/6 with trample and first strike

It's worth noting that the p/t switch used to be timestamped and would only swap the p/t that happened before the swap.

Well that sucks... at least I know now. Layers are stupid.

They're actually a lot better than they used to be.

It looks like that's still the case. I dug into the CR to try and understand better, and found this;

So doesn't that mean that the swap should NOT influence the Sure Strike?

Layers are indeed stupid... how long ago was it that the timestamp would have allowed for the interactions I mentioned to take place?

It just seems odd from a rules perspective to have something like a P/T boost spell (Invert) to resolve, but continue to check well after the fact. That resolving Invert and then casting 20 spells could result in like a 3/22+ is crazy...

Nope. That is saying within a layer or sublayer, then it's timestamp order. But the power/toughness swap is a different (last) sublayer than the adjusting power/toughness via effects or counters. So the p/t swap is always last.

And really, it's for the better. This way you know you just calculate the power and toughness, then swap them if needed. Instead of having to determine if an effect that altered p/t had an earlier timestamp or not (which can then get into arguments about what happened in what order -- I've not seen arguments, but I've certainly seen games stop as people try to remember what order it all happened in.) This way keeps the game moving when you do a p/t swap.

Example of timestamp mattering: Diminish is cast on a creature, and Creeperhulk is activated targeting the same creature.

Those both set power & toughness to a specific value, which is applied in the same sublayer:


In 7b, we apply them in timestamp order, so whichever effect was most recent "wins", making it either a 1/1 or a 5/5. (Also it gains trample either way in layer 6.)

Either way this still applies before we might increase power/toughness in layer 7c, such as with +1/+1 counters.

This is a much better way of explaining it than the way I did it

Re: Timestamping - that only applies, as you noted, within a layer or sublayer. Since the p/t altering is in a different sublayer to the p/t switching, timestamps for those two different effects never matter.

Historic addendum:
If you look at older cards (read: really old ones, like About Face), you will find this oddity of P/T-Switching spelled out. I guess in these "ancient times" of Magic they had to make a decision how to handle later buffs.

I love layers questions but mostly because I committed a fair amount of effort learning about them. The answer given is correct, you apply the effects in the order set out in CR613.

For a handy reference guide, I recommend the phone app - MTG: Judge Core App. It is designed by a group of judges and can be used offline.