1. The slum is NOT an upgrade to the house.
For kingdom: hut (25) -> house (40) -> villa (60)
For empire: hovel (30) -> house (50)
then there is a separate improvement called Slum, that takes up FOUR TILES, uses two food and allows 120 people to live in the building
Kingdom has better possibilities here, empire has worse.
2. The great scimitar makes the wearer lose both dodge (-3) and accuracy (-3). In contrast, the claymore makes the user gain accuracy (+6).
I still agree with you, that the great scimitar is overpowered. However, the difference is not as large as you think.
3. Almost everything that empire gets from their exploration tree, kingdom gets from their civ tree. Empire also needs to spend more time researching exploration to unlock the resources that spawn from the tree (6 vs 4)
4.
Empire monument: 0.5 prestige, 2 gold upkeep, 5 turns to build, 10 specialists
Kingdom: Inn, 1 prestige, 2 gold upkeep, 10 turns to build, 10 specialists
Pub, 1.5 prestige, 4 gold upkeep, 15 turns to build, 10 specialists (requires tech)
If a kingdom city has built a pub and an inn, the empire city would need to build 5 monuments to get the same prestige. And they'd lose 4 gold per turn and 30 specialists more to get that, compared to kingdom.
The kingdom is, again, ahead in this category.
If you're looking at something truly unbalanced, look at how quickly empire can get to sages tech, unlocking their version of the library. That alone should be reason to play empire.