Will this version support 1366x768 screen resolutions?
This is the biggest problem I have at the moment.
I wish I could play on my notebook.
It depends heavily on the specifics of the device but if you have a dedicated GPU in your notebook take a look into DSR/VSR to simulate a higher resolution and then have it downsampled to whatever your display resolution is. (DSR for Nvidia GPUs, VSR for AMD GPUs)
However even if you have a dedicated GPU there's a good chance you have a Intel CPU with a iGPU and it will be the CPU (and as such the Intel iGPU) that's connected directly to the display rather than the dedicated GPU so DSR/VSR won't be a option, in which case it may be worth looking into 3rd party downsampling software solutions, there's a few out there.
I believe if you have a AMD CPU regardless to if you have a dedicated GPU or are using a iGPU on board the CPU chip then VSR should be available to you. So it's only if you're using a Intel CPU that you would run into issues and need to find 3rd party software work arounds as I don't believe Intel have yet to bother offering the functionality (And show no interest in ever doing so)
Most applications have zero issues with downsampling higher resolutions... Nvidias solution is a bit more temperamental than AMDs when using it outside of fullscreen applications from my experience switching between the two at various points, as Nvidia don't officially recommend DSR be used for desktop/windowed use.
How Aurora would react to having 'fake resolutions' fed to it that are actually downsampled to native resolution would certainly be something interesting for Steve to look into before C# releases if he hasn't already though.... as in terms of what the software thinks, it will think you're using whatever resolution you have DSR/VSR/Misc set to, as even the OS thinks you're running at say 1440p when actually it's 1440p downsampled to 720p or similar.