Will any (recent) Linux terminal emulator allow me to save my session (tabs, history, etc)?

this isn't exactly what you asked for but tmux has such capabilities.

Just make sure to install the tmux-resurrect plugin along with it, which allows for restoring the tmux environment (windows, splits, and certain running programs) after a restart.

On the bright side, tmux will work with all

Here is an xfce terminal fork, with possibility to save/restore session just from menu: https://github.com/repu1sion/xfce4-terminal

https://github.com/Eugeny/terminus saves open tabs & remembers the correct directory for each

The excellent Guake Terminal allows you to do this, there is an option to remember your set of tabs between restarts.

It's currently listed as an 'experimental' feature in the settings, but after using this feature for a few months, I've never had a problem.