Compiling is usually painfully slow, sometimes 5-10 minutes.
I keep seeing errors about not being able to reach language server.
I don't want anything I do to wind up in the cloud.
I'm hoping the answer to all three of these is to configure the IDE to work offline. I don't mind a connection for library install and updates, but not for every compile, or god knows when else.
I'm sorry to inform you that it won't help with the first two, and is not necessary for the third.
Arduino IDE only communicates over the Internet at specific events, which you trigger yourself:
It checks for the availability of updates on startup
It downloads the package index files if you add a new URL to the Additional Boards Manager URLs preference
It downloads the Arduino boards platform files if you install or update a platform via Boards Manager
It downloads the library files if you install or update a library via Library Manager
It is unlikely that you would doing any of these things at the same time while the sketch is compiling, so the IDE's ability to access the Internet is having absolutely no effect on the compilation time.
Despite the "server" in the name, the language server is not something that is accessed over the Internet. This is a program that is running on your computer that doesn't use the Internet in any way.
It is true that Arduino IDE has the capability to save sketches to the user's Arduino Cloud account:
However, it will only do that if you take action to do so. It never automatically sends your work to the cloud. So if you find the Arduino Cloud sketchbook integration useful, then you can use it, but if you don't think it is useful then simply don't use it.
If you want to completely disable the Arduino Cloud sketchbook integration, you can do that through Arduino IDE's advanced settings:
Press the Ctrl+Shift+P keyboard shortcut (Command+Shift+P for macOS users) to open the "Command Palette".
A menu will appear on the editor toolbar:
Select the "Preferences: Open Settings (UI)" command from the menu. ⓘ You can scroll down through the list of commands to find it, or type the command name in the field.
A "Settings" tab will open in the Arduino IDE main panel.
Type arduino.cloud.enabled in the "Search Settings" field of the "Settings" tab.
Uncheck the box under the "Arduino › Cloud: Enabled" setting.
Close the Preferences tab by clicking its X icon.
To repeat what I wrote above, there is absolutely no Internet communication involved in compiling.
Slow compile: are you using ESP32? You can search on this forum for a few (long) threads about "excessive" build times. When you break it down, it "makes sense", and "not just you", although that doesn't actually solve the problem. Aside from better hardware, there's no much practical you can do, aside from:
Linux is faster on the same hardware
If you're using Windows and have the disk space, repartition some as a Dev Drive and work from there
In any case, you can post more specifics: board, OS, etc for your ten-minute compiles.