I first heard of the Junon alarm method about 6 years ago when I last played FF7. It was in a YouTube comment (back when Google+ didn't exist). I've always hit level 99 in my 6th leveling ground, upon obtaining the Tiny Bronco. Upon getting the buggy, I went for trying this method out. It's just as effective as my 6th leveling ground and it's accessible much earlier.
Due to the alarm being rather annoying, consider shifting the balance to the left. This nearly mutes the game's audio but keeps my voice. I stated right, but it was a mistake.
When you first start this, you must have the beta enemy skill or it'll be too difficult to handle. You get it from the midgar zolem. To make it easier to get, give everyone sadness status to absorb 25% of the damage you'd get and poison the snake with bio. Once poisoned, have everyone but the one with enemy skill attack it with normal attacks or weak spells. When you get "limit break" appearing when it's not your characters, it's just about to use beta. Let it use it. It'll do about 800 damage to everyone (400 to the one with elemental+fire for the materia in the armor) if you have sadness status, about 1050 without. When beta is used, have your main healer cast cure 2 on all and unleash the limit breaks you've developed right after to finish off the midgar zolem. Also, the "long range" materia will be especially useful.
Now, if you're below level 35, use beta once to significantly weaken the enemies and finish them off with regular attacks or matra magic. Heal and/or revive as needed. With sadness status, you'll usually get about 200 damage per hit (150 with sadness) so these enemies hit hard but they're easily tolerable if you're prepared.
For death machine, wait for it to attack first before attacking it. If you do, you'll only get attacked once as opposed to up to 4 times for potentially 400 damage a hit (300 with sadness). After the initial 3 attacks from everyone, dedicate 2 characters to attacking and the other one to healing.
For slalom, just use beta right away to make things much faster. If beta doesn't finish them off, just use regular attacks. You shouldn't need to heal and ignore the poison and blind status effects.
Once you're above level 40, don't bother with using beta on death machine any more and just attack it physically instead. From 35 to 40, usage is up to you. When you can consistently defeat the slaloms in 2 hits, use regular attacks instead of beta only when you get 2 of them. When 2 characters can regularly defeat them in 1 hit, only use beta if the character with it is the first to act and only if you get 3 slaloms.
Happy grinding to level 99 so early!