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I have a print composer template saved, but when I open it in a new project I can't see a way to get the map to update to the new project. It just stays blank.
Is there something like the legend where you can auto-update it or link it to the current project or do I need to add the map in manually every time?
I assume you're missing the button shown in the image below. You can find it in by clicking your map and navigating to item properties dialog. There in the uppermost row you can find a set of buttons with which to alter map extent. The button you're looking for is called Set Map Extent to Match Main Canvas Extent. This of course is a manual solution and it requires you to have the extent you wish for in your main canvas.
Please note that I don't unfortunately have the newest QGIS 3.10.x installed on this computer so the screenshot below was taken with QGIS 3.6 and thus some changes might have occurred.
May be I didn't understand all your expectations, but you can check this:
How to import a Composer Template into QGIS Print Composer?
In addition for your problematic about auto-updating the legend, yes there a way:
In your legend option, you have to check the option of auto update and filtering the entities of your legend with the funnel.
