PowerShell and Tee

1. My-Command 2>&1 | Tee-Object 'myfile.log'. See Get-Help about_Redirection.
2. You should be trapping errors, not relying on Ctrl+C. See Get-Help about_Try_Catch_Finally. Is the command you're running an external program or a script?
3. As I understand it, typically string objects are not sent down the pipe until an end-of-line character is reached. The reason is quite simple: if it didn't do this, partial (read: incomplete) strings would go down the pipe. Tee might handle partial strings fine, but other cmdlets like ForEach-Object or Select-Object certainly would not. Note that Get-Content has a special switch -ReadCount which somewhat overrides this behavior, and it will seriously mess with a Select-Object -Skip/-First/-Last/-Unique command further down the pipe.

It very well may be that external programs you're running aren't going to obey the conventions that PowerShell is expecting. Tee, for example, is properly called Tee-Object, which should tell you the kinds of things it's good at working with. In that case, you may be further along the line to get tee.exe from GNU Win32 Utils or MSYS which are designed for immediately forwarding content.