The AI in this game cheats.

We had this discussions many times. And no, the AI does not cheat. It doesn't do anything you couldn't. It simply knows better than you how to achieve these things.

One key is diplomacy. The AIs don't research everything by them self. When a game is finished, check the stats screen and you can see that they got most of their techs via diplomacy.

"The simple truth is, I expand incredibly quickly, set my spending cap all the way up, crack out piles of colonizers, my techs are researched in only 1-5 turns, and by the the time I've finished researching "Corvette" or "Shields," my opponents are cracking out Battleships and have significantly higher populations."

I'm not sure if AI has any bonuses on normal difficulty level - shouldn't have iirc from my first few games - note that speed of your research/production depends on:

1) speed of spending your revenue ( higher = quicker production )
2) your spending adjustment ( hom many percents of your income you're spending on research, social and military production)
3) total number of research points from every planet ( only research )

It means that you can have more planets and still be outspeeded by AI players with less amount of planets but better developed/with higher PQ.

If it's going about military developmnet - check out technology tree and exchange technologies with other races ( it's very important ) - AI is doing exacly the same - note that in some moment of the game, you will start to have more techs that AI players haven't researched yet - it's because they're researching some military techs while leaving social projects ( like Battleships or Dreadnaught ) - still if you don't have anything more powerful than corvettes when enemy is cutting space with battleships engines, your development is too slow - don't worry I'm sure you will learn quickly.

"...and have significantly higher populations."

Check their PQ - AI tries to be as efficient as possible - it's researching projects boosting up PQ , income ( "Trade" tech early in game - you should get it also ASAP ) and production quickly - also AI is aware of yellow stars positions thus probability that alien race will grab better planets than you is significantly greater unless you will get speed bonus.

"They also have the ability
to launch waves of these ships at me and the expendable population in transports. PLUS they each have around 10 more techs than me a piece -- and they only have an average of 5-9 planets while I have around a dozen or more."

You can boost up your own production by getting proper political party ( for ex. industrialists - my favoured ) + getting military, social and research bonuses instead of for ex. PQ or speed bonus. - remember also about doing proper choices when chosing techs to research - fusion power plants, manufacturing centers, manufacturing capital, antimatter power plants etc. will boost up your production efficiently.

Moreover, you can build starbases in sectors where your systems are - by adding modules via contructors, you can boost up your production up to 120% from one starbase ( building additional one will give you another boost because bonuses from starbases stack so 2 starbases with fully upgraded production modules give you 2*120=240% of production bonus - it works also for trade, culture influence modules etc. )

"So what is going on? Is this as intended? Should I cheat as well--because "Intelligence" means to have easy access to virtually everything warmongering? "

AI is not cheating in this game - it has some serious bonuses to PQ, research, social and military production + free techs at the beginning ONLY on higher difficulty levels - on normal diff. level only bonus AI gets is probably ability to see yellow stars ( dunno about free techs, probably not or few only ) but it has been done to balance human intellect, that cannot be outmatched afterall ( after few games you will find beating AI on normal difficulty level too easy - now I'm playing on masochistic and still AI, after initial phase of game, is quite an easy prey.