Can you play 1080p on a 1440p monitor?

I don't see why not.

There is really no reason to call it 1080p, all displays are progressive rather than interlaced these days.

This section is filled with stupid people, you shouldn't trust ANY of them, especially if they're so dumb as to claim that there is "lag" for upscaling. Worse is that they're so stupid as they don't know the difference between what the monitor does (yes, some "smart" (aka stupid) monitors have weak graphics cards inside to do bad post processing) and what the GPU does.

They probably have NEVER opened their GPU control panel, and are the type of people who should never be trusted.

You definitely can, however if you are fussy won't look that great even compared to a native 1080p monitor since most of these monitors don't do the best job of up-scaling. It's always best to use the native resolution.

Yes, you can. With 720p or 360p, whatever, just configure the settings inside the game

You can but it's not recommended. Using a non-native resolution just means that the monitor itself ends up upscaling the image to what its true resolution is.

Use the native resolution wherever possible, or go for an exact multiple (such as 720p) so that the image scales evenly.

Of course, but depending upon processing and monitor settings, frames per second lower and lag could be higher than matching native resolutions.

Per comment, what I am stating is that 1080p signal game settings output on a 1080p display is faster and less lag than on a 1440p display.