1. No, there is no treaty for "stay off my land or I'll go to war with you." You can either tell them to get out every time they enter your territory (by speaking to them in the diplomatic dialogue window), or you can ignore it. Or you can do what I usually do, and go into a permanent state of war with several of your neighbors and ... remove ... the trespassers.
2. The option to raze a city is not supposed to be available to players until you've occupied a city for five or more turns. It should either display next to the Build/Train/etc buttons on the selected city menu, or be in the drop-down menu accessible through the button that looks like a couple of arrows (>>>). Unfortunately, the AI doesn't always follow this rule. Also be aware that razing a city often prevents resettlement of that tile, so it might be better to just accept whatever the AI built. Plus, rebuilding cities can take a long time, both for the city to grow and for the structures to be built.
3. Auto-update takes a unit you've already trained and upgrades its weapons or armor (depending on which you told it to) to what the computer believes is the best current weapons and armor of the type present on that unit. Be aware that you might not agree with the computer on what is best (e.g., the computer will switch out Monk's Robes for Aegis Robes and probably Monk's Staffs for Mauls, and might upgrade your daggers to broadswords when you really wanted shortswords, or other things like that). Upgrading also costs some of your resource stockpile, and so isn't always worthwhile, especially if you can train units with more members than the unit to be upgraded has. As far as I know, the design itself will never update, but if you tell the computer that the design should use the 'best available X' where X is a type of weapon or armor (it's the icon with the up arrow on it in the design screen), the design should be produced with what the computer considers to be your best X. Note that weapons are divided by damage type, range, and number of hands, so axes can upgrade to swords, and daggers might upgrade to axes. Also, as far as I know, the computer considers 'best' to be 'highest damage', regardless of any other benefits or penalties.
4. Try deselecting "autoselect next unit" in the options menu. I'm not sure if it will work, but I think it might.
5. No idea.
6. No idea.
7. As far as I know, you have to go into the City Details screen (either click the circular icon in the lower left corner when a city is selected, or click the 'Details' button when a city is selected) and inspect each city parameter (production, unrest, food, etc) to see what is affecting each of them. It's inconvenient, but it's the only way I know of to see curses an enemy has applied to your city.
For your own enchantments, usually there should be an icon displayed above the city name on the world map (assuming you are zoomed in to see figures and things like that), or there should be an icon in the upper left corner of the city details window. There is also a currently active spells window (I don't know the exact name) in the empire overview, which shows you all of the maintainable spells that you have cast on your champions and (I think) cities. You can dispel such enchantments from both the city details screen and the empire overview screen.