Hi and welcome to the forums ^^ I know I'm not the first one to comment here, but I'll throw in my 2 cents as well.
First things first. Never, and I mean never, put any of the AI homeworlds or coreworlds (the planets bordering the homeworlds) on alert, unless you are positive that you can destroy the AI homeworld and plan to attack it very soon. You put a planet on alert when you have military ships flying through any of it's neighbours or when you destroyed/captured any of it's neighbours. Putting a high mark world on alert can have serious consequenses over time and putting the core/homeworlds on alert is almost always a very bad thing to do (unless you actually plan to strike of course).
In case you don't already know, coreworlds appear to be regular mk IV worlds, but they're not. It's very possible that you scouted a coreworld, but not a homeworld yet, and accidentally put it on alert. That happened to me once or twice in the past. Not only do coreworlds have a lot of mk V ships, coreworlds also always have 2 warp gates on them. This is how you can tell them apart from regular mk IV worlds.
If you, for some reason, have to capture a coreworld or a planet bordering one, build a warp jammer command station. Not only do they prevent waves from being launched against your planet, they also trick the AI into thinking that the planet you captured is still one of theirs, thus preventing the alert.
I think the reason you see so many carriers with lots of ships is because a coreworld is on permanent alert. When a planet is on alert, it gets reinforcements over time. Tough planets get more and stronger reinforcements. Eventually all the guard posts will be crowded with ships and that's when reinforcements start to leave their guard posts and become threat (stalking the wormholes in "active" mode). When there are too many threat ships on a world, the AI starts to put them in carriers which they eventually send to your planets. This could happen very quick when we're talking about core and homeworlds. Also with the new threat behaviour, the stalking ships may decide to attack any of your planets quite fast. Keeping threat numbers low is very important.
That covers one part of the stuff I wanted to say. As for the AIP issue:
My own personal playstyle is all about high AIP games. I'm not skipping 3-4 worlds to take one planet and then skipping 3-4 worlds again. I scout for a nice target (ARS, Adv Factory, Golems, Dyson Sphere, SCL, etc) and once I've found one, I move in a straight line from my nearest world to the target, almost always capturing everything along the way. My games practically never end below 500-600 AIP, yet I manage to beat most of them, including difficulty 9/9 games.
Gravity turrets and tractor turrets are your friends. They should be a part of your basic defenses at ALL times. Always neuter bordering AI worlds wherever you go (killing warp gates and all guard posts except the wormhole ones, and leave the command station alive!) and just leave 1 single bordering warp gate alive. This way you can control where all the regular waves will come from and you can build your defenses accordingly. When you are absolutely certain that all regular waves can come from only one single wormhole, build Heavy Beam Cannons and Lightning Turrets near it, preferably as close to your command station as possible but still in range of the hostile wormhole. Don't forget the grav and tractor turrets either of course. Use your other attack turrets to defend the rest of your empire, placing them the exact same way as you would with HBC's and lightning turrets.
There are many different ways to succesfully defend your planets, but this is definitely one of them. I use it every time and it works every time. I hardly ever have to unlock mk II attack turrets (except for the HBC's and lightnings, which often even go mk III in my games). Just make sure you keep enemy numbers low on bordering planets, even AFTER you already neutered them. High mk worlds often require attention or they will reinforce too much. Don't let those numbers grow bigger than you think you can handle.
Making progress might be going slow this way when you have to do so many things to keep the AI under control, but it brings you much closer to a possible victory.
As for attacking, don't suicide your whole fleet on a planet if you think they won't be able to handle it. Retreat and rebuild. Even just killing a single guard post or ion cannon or any other structure before you have to retreat is still considered making progress.
Trying to keep the AIP as low as possible is not that important. What's important is that you find the right balance between keeping AIP low and not letting the AIP get higher than you can handle.
I know this is a lot to swallow, but I just want to help =) If I said anything that doesn't make sense to you, but you do want to know what the hell I'm blabbing about, feel free to ask me and I will try to explain it in a different way.
Good luck with getting your ass kicked stomping the AI and earn that 7/7 victory =)