So by Day 7 I've ran out of money? Game keeps telling me that the cost is more than I can pay or something like that every time I click OK.

Time to restore?

BTW, what's the deal with not having any production at all? At the beginning of the game there was a production plant in Sector 60, but now it's gone? WTH

It sounds like you didnt get the micro purifier back on day 1.
Did you assign workers to work on the trees? That's the only way to get money.

Yeah, I did.

Meh, I restarted the game. Let's see how it goes the second time around.

Your team is probably too expensive. Either you have expensive mercs, too many mercs or the combination of the 2.

Some startup teams have been mentioned here in the past, so you might want to do some searching.
A good start team which is affordable:
Ivan
Fidel (if he refuses to do something because he's busy, move the gun from this hand)
Speck (mule on day 1, repairs for the rest of the tour)
Wolf or Ice

You could replace Wolf or Ice with Jimmy or add him for his lockpicking skills.
You could also add a cheap doctor just for the doctor and FA skills. He/she might not be usable in combat, but you could use him/her just for doc/FA skills and as a mule.

Expand your team once your income grows.

This is probably a redundant tip since I think it would be very hard to reach Day 7 without hiring any, but you need to hire tappers to make money out of the trees you have captured. I recall the first time I played this game it took me a couple in game days before I figured that out.

But now that I think about it, what may have happened is, at some point in the campaign, Jack will only be able to get $400 worth out of a processed tree, instead of the $500 you got at the beginning of the game. That event is scripted in the campaign, there's nothing you can do about it. So, if you were low on money and your expenses were pretty close to your income, and that event was triggered, your budget would go in the red and you'd be unable to afford another day without firing some mercs or guards.