If you are exporting, it is best to just use the “Export” function as it downloads and exports in a single step. The “download” function is primarily used when back-filling or verifying existing data. You can obviously use the “Download” function if you want to merely pre-download the data to your PC for subsequent use at a later time.
And yes, you can copy the FXT files to the Data Folder of another MT4 instance.
Once the data is downloaded, it will reside on your disk drive and will not need to be downloaded from the internet again. There are more details on the download behaviour in the manual.
If you want to change your MT4 Export Settings (i.e. FXT information), then you can do this in the “MT4 Settings” window when you export MT4 data. You can also save your settings as “default” so it will use the settings on the next export. This feature is also detailed in the manual.