For ZvZ Hard AI mode, you need to go roach fast because roach owns zerglings pretty hard. The hard mode will usually send a wave of roach + hydralisk and if you only have zerglings and spinecrawles defending, you are on the loosing end. You could also try a 6pool + drone to build a spine crawler in their base, use your lings to def the crawler till its done and u should win (this also trains your micro skills of your lings to eat up as much of their lings, you have 6-8 to defend the spinecrawler till its ready. The AI generally will be able to wipe your lings because of their micro)

I find that for zerg, you shouldn't even let the opportunity of your opponent to proxy inside your base. Your overlord should have sufficient time to cover the ramp for you to monitor any incoming scv/probes. Its only the proxy in your nearby expansion that you may not have scouted that would cause some problem to your mid-late game.

In the event that a proxy pylon had been established and cannons are warping in, your next best move is to kill the probe first, tech to roaches. place a pair of zergling at the ramp because your opponent will send in another probe to rebuild. mass up around 6-7 roaches and go straight for the pylons, you have set your opponent back as much as he set you back in terms of economy.

If he placed 4 cannons 2 pylons, you have won the economy race, he spent 600minerals on useless structures and further 200 on pylons, and another 200 to replace the destroyed pylons. 1000minerals gone! not to add in the forge cost. Your immediate counter of around 10 roach will be sufficient to break into his base.

You can definitely learn a few tricks by playing more AI games.

At one game of Hard AI mode against protoss, I was practicing the fastest way to mass roach and attack. I was met by a fast immortal build from AI and it countered my roaches.

In any usual Hard mode, the roaches will be able to take out their zealots and stalkers with ease, but against immortal its pure suicide. This is then where you can learn from the AI how to go fast immortal and yet have certain units to defend early. There's a few tricks up their sleeves that you can learn from.

In another game, I prepped for an early immortal so I went with 50% lings and 50% roaches for an attack. only to be met by an early colossus that wiped my lings and roaches. FML .. lol

And I tried these builds in team leagues, we were able to win when my partner just needed to mass tier 1 units and def for us, and once i get my early immortal or colossus out we start pushing. Other team went "omfg an immortal / colossus that quick?"

Btw something cool about zerg is that you can infestor control a probe, build nexus and start your own protoss production. You can then chrono boost your zerg structures too. Silly stuff but tried it yesterday against an AI for fun and to get some achievements like 'zerglots'. Chronoboosted spawn larvae is awesome.

I have not tested in 2v2 for sharing units and chronoboosting yet. But i feel it may be viable if you are controlling shared unit.

But for sure you can chrono boost your own zerg structure if you build infestor to mindcontrol the probe and develop your own nexus. In FFA this is often one of the best tactic when you nydus rush someone and they leave the game after getting wiped significantly. Dont kill all of their units. Go for the mindcontrol and take over their builders to build their class base. That's where the fun beings.

for T to tech to thors, they can do it without giving much breather for the Z player to freely expand. the marines and hellions combo with scv as meatshield and mech repairs will be able to harass very effectively not to mention aggressive bunker forcing Z to be confined if the Z lacks the force to apply pressure. all these while the T player can still be producing the same units and completing the armory and tech labs for more mech push like thor. a single early thor with 4-5scv repairing is pretty deadly.