You can take cities at any time, you just have to be prepared for losses.
I tried playing Great Plains for the first time today, decided to start off on Monarch. I tried to take the English capital @ 40% culture with 6 CR2 swords, a couple pikes, an axe, and a handful of skirmishers. I'd already taken one city, which was really the only goal of the war...Christian Holy city, ended up with an Academy/Shrine in it as well, relieved culture pressure on one of my cities, and gained a couple resources. I came up one troop short of the capital after my 6CR swords all lost with 72-98% odds, leaving a .3 archer in the city.
The next turn, I had 3 very wounded troops, and they had moved 3 healthy horse archers into the city along with the archer who promoted. I had to sue for peace without the city or the tech I wanted, due to the vendictive RNG gods, so I *only* got the one city.
So...you can make some huge gains by attacking early. But you need to be prepared to take the losses that will inevitably come along with it. I was in a position where I really had nothing to build but military anyway in my core cities, so it didn't matter much...and the troops were costing me money. The war gave me more land and resources so I could grow, crippled my oppenent, and allowed me to raise science by lowering unit cost. And when the city I did capture came out of resistance, the shrine/academy boosted me into the lead.
I actually built the Pyramids as well, which I never do, so it's not like I sold out for early war either. I think I had 6 cities, was doing well in tech, etc...
You definitely don't need cats. But they help. Especially with the horribly high WW you get for losing attackers during war. Part of why I am very hesitant to suicide cats anymore.