Economic Problems

It probably would help but there's no single answer to your problem. You probably need to colonize better as well as be a little more indirect in your first few encounters with the AI.

Clearly as you move up in difficulty you are going to find that the AI gets stronger and it's harder for the human player to keep up. When that happens you still should look for ways to try and keep up however you also need to begin to rely less on brute force and more on subterfuge to get what you want.

Instead of declaring war directly try paying someone else to go to war with your intended target first. If you pick the right opponent it can be surprisingly cheap to do so. Then you can sit back a bit and get yourself set up to swoop in after they've been softened up for you. Do this a few times and you'll soon be strong enough to fight them on a more even keel.

Declaring war on a seemingly defenceless race is never a good idea unless you have enough transports to take all of their most productive worlds.

See above, getting someone else to soften them up is great and means you can see whether they actually have the capacity to fight back.  Not to mention you're potentially screwing up relations (ending trade and treaties) between the aggressor and your intended target for some time.

If you absolutely have to attack but can't spare the time to research medium hulls yourself, get ahead in logistics.  It will allow you to band those fighters together and kill frigates before they get fleeted up - Terrans start with a Logistics score of 6 and move up to 9 upon finishing Basic Logistics research, which isn't enough to fleet two frigates.

Having the best industry does have advantages when fighting a war but unless you need every planet to be producing warships, it's best to specialise some of them by demolishing manufacturing facilities on the ones furthest from the action and building trade centres or something instead.  Not only does this mean that you have more taxes coming in, but it means your military spending is going where you need it most.  Most of the time you're going to capture planets with factories and labs, and these can produce fresh ships closer to where the fighting is going on.

I found - as I moved up the difficulty levels - that I had smaller and smaller "windows" that I had to get through.

That is, on lower levels, I could colony rush for a while, over-extend a bit, and still recover, perhaps with treaties and playing off one AI against another.

On higher levels, I learned to be more disciplined when I stopped the colony rush so as not to over-extend, to focus my research on specific goals, and to get future enemies into early wars.  The Drath, in particular, tend to get races to declare war on the human player, so it pays to get most of the galaxy at war with the Drath early.  That means getting a diplomacy edge early, even if one cannot sustain it.

I found it important to get the early civ-wide bonus techs done and run to Trade simply for the economic capital, then not long after to Democracy to get the political capital.  Those two help buy time in income terms.

Well, you had a military but apparently no economy. Bad that is. Next time try to play it exactly the other way round. Be strong on econ - assuming concentrating on pop first. This will give you more options as to what decide once problems will occur.

In short, without money your're boned.