I get copper about 25% of the time. On the other hand, about 5% of the time; no copper, no iron, no horses. No sense crying over spilled milk: Archery then Theocracy and Vasselage then it's cats or trebs + promoted longbows. As TMIT observed, longbows are not the best offensive troops in the game but they seriously chew up AI counterattacks and if you can pillage the enemy's horse and iron resources then camp a few longbows on them you can win the war through attrition. AI seems unusually stupid about recapturing pillaged resources. Moreover, your longbowmen perched on the iron resource are likely getting a 25% hills defensive bonus which makes their job all the easier.
A good pillaging longbow camped on an iron hill will have +25%(hills) +25% (archery unit in hills) +50% (guerrilla II) and will defend at strength 12 before counter promos or fortification.
Between that and mobility, such a setup can be lethal, especially to humans who don't have the AI bonuses to stock each city with 3+ archers a piece and still field something else. Attacking those things on hills is like attacking grenadiers on the open field...in 200 AD X_X. They of course lose all that if they are attacking instead but for the choke role they're devastating.
On difficulties where the AI hasn't chopped everything yet (or against foolish humans or just very large amounts of land), woodsman II swords also enjoy an incredible boost in the forest. Again, you're looking at +50% (terrain) and + 50% (woodsman II), making them very obnoxious to attack there. A combat I shock axe attacking such a unit is 5+5(.1)=5.5 (axe) vs 6+6(1-.75)=7.5 (sword)! Anything else attacking will do miserably. Swords are a little better on city attack and cheaper than longbows, but unlike hills forests can be chopped.
The above 2 examples are even more ridiculous if you're on hill forests/jungle.
In SP your siege is usually doing the heavy lifting after very early rushes anyway. If you're playing below immortal in tight quarters it is actually viable to warrior rush non-PRO civs, too. Actually, I've been experimenting with it on immortal/normal and it does seem to work there as long as they don't have capitol BFC copper (you hit with about 16 warriors around 2200 BC, build order worker ----> rax -----> warriors).
On the topic of copper, IMO it is not the best rush resource. That merit goes to horses. While weaker directly, chariots move faster (negating some whips and AI garrison shifts, as well as basic production if you take cities quickly) and cost fewer hammers. They are also frequently revealed sooner, since having livestock is STRONG incentive to research AH as pigs/cow especially are some of the best tiles in the entire game. If there is horse nearby a 2000-1800 BC strike with 8+ is lethal.
I've never double-rushed (aka killing 2 civs outright) with axes on immortal, though I've tried. I've done it multiple times with regular chariots.