I finally decided to man up and try Hardcore difficulty to get the achievement and unlock Insanity, but I'm having trouble. I'm on Noveria and I reached the part where you finally leave the Mako and enter Peak 15. My team gets attacked by a Geth Juggernaut, which isn't too bad because once you get his shields down he dies in a few shots. But the Krogan are giving me lots of trouble because once they start charging, they end up running right up to us and killing us with melee attacks. My team consists of Ashley and Tali, which may seem like an awful combination because Tali is weak sauce, but I'm also going for the achievements for completing the majority of the game with those two characters. Are there any tips you guys have to make this easier? Oh, and I'm playing as an Adept with the Assault Rifle bonus skill, but those damn Krogan....

Use Stasis/Lift/Throw for the Krogan, if you take the Bastion specialization you can continue to damage enemies while they're in Stasis which is very good for Krogan/Armatures/Bosses (Maxed out Lift can also immobilize the Geth Colossi). In addition to the physics based powers make liberal use of Warp and Neural Shock to counteract the Krogan regen.
 
Tali can be good if you max Decryption/Hacking/Overload, disabling a tougher enemy's ability to shoot can be very good. In a few of the on planet raids you'll have close quarters and rocket drones, anything you can do to stop them from shooting will save your life - Stasis/Hacking/Overload.
 
I just ran through the game using Adept w/ Assault Rifle, and if you can help it just don't level up any combat points; by the time you're level 40+ the guns will be accurate enough that you won't need Marksman/Overkill and would be better served maxing out all of your Biotic abilities (Leave Charm/Intimidate alone, as they can be level up via actions).

Switch who you're controlling a lot, you really need to make use of everyone's powers as much as you can, and try to use a lot of Tech because it weakens their sheilds and makes them more vulnerable to Biotics.

For that situation you really need a few lifts or throws. Tali and Ashley may not cut it, although I would probably start by overloading their weapons and casting immunity (or whatever the ability is when you get boosted health) on Ashley. 
For the Krogan in particular you really need to kill them before they boost their health to insane levels. 
  
I don't if this would work, but try luring the juggernaut out of the range of the krogan. If you can break the fight up into chunks you'll do much better.

" @ApolloJ85: The Juggernaut usually comes at us first, and he's relatively easy to defeat. After that, a Krogan will charge in and by the time he's dead both Tali and Ashley will usually be dead. So then I'm by myself to fight off the remaining 2-3 Krogan, one of which is wielding a sniper rifle that kills me in one shot. "

I did hardcore at level ~ 55ish and insane at level 60.  
Tips what I would reccomend is to have the best gear for your squad and turn the squad power usage to limited. 
Using limited allows you to exactly pinpoint the skills they use. Under normal scenarios the squad is pretty stupid, using skills at random that don't make sence. 
For you Singularity is golden. Not only does it attract a lot of enemies into one location, but allows your squad to unload powers into the 'cluster' of enemies. 
Lift is also very good, as it allows you to pick up a group of enemies and throw them away if you so choose (or unload powers). 
While playing as an Adept, don't rush in, your squad are tanks and you are the crowd control. 

Chemical rounds got me through the harder difficulties. Use lift on an enemy and then pepper them with toxic rounds as they float through the air.

play as a soldier, be at least level 40 when you start. Just change the difficulty before you leave Eden Prime.
 
I can't imagine playing on hardcore or insane without immunity.

I turned the Squad Power Usage to Disabled so I have more control over it. I was able to get past the Krogan and Juggernaut fight with ease, but I'm stuck once more. I'm at the part where you trigger the purge to kill all the Rachni, and you have to make it to the door before time expires and you blow up. How am I supposed to do this? When I tried it earlier today, I killed most of the Rachni but was forced to make a sprint for the door because I only had about 15 seconds. I died. Is there an easier way to do this? I'm not sure I could kill all the Rachni in time, and there's no way I can run through there.

I never control my party members.  (Yes I beat it on Hardcore from lvl 1)
 
Equip your guns with the mod that makes them do extra damage against organics.  Use Push/Lift/Singularity to keep them off of you.  Also don't overlook Warp.  As it increases your damage against the person you use it on.  Against Rachni use Chryo grenade mods.  I beat that part by casting Singularity in the middle of the room.  Then Barrier to raise my shields really high.  Then made a run for it.  I still barely made it.  That part is a pain in the ass but just keep trying.