I have had this tube of MX4 since 2013, it's served me well, countless rebuilds of my computers, CPUs and GPUs alike, home servers, gaming computers, laptops, games consoles, pi4, I used this on everything. You served me well.
8th June 2013 to 4th November 2023.
No, but you do want to occasionally rotate them around. They will separate over long periods of time.
If you have some expensive stuff that has been untouched for like a year, plunge it all out on a cleaned piece of glass, very, very, thoroughly stir it back up, and scoop/smash it back into the back side of the tube and then let it sit upright somewhere for about a month so all the air works itself back out to one end before using it again.
This really depends on how the particular paste is made. I've never seen MX-4 (carbon) or NT-H2 (ceramic) de-emulsify.
It's not something you'd visually notice all that well. Just that the carbon particles (using mx-4 as an example here) will drift downwards instead of being equally dispersed in the paste. That can leave your solution you apply having too little carbon to do its job as well as it should.
No pastes are just liquid. They're all super finely ground solids suspended in a liquid, and the solids never weigh the exact same amount as a liquid, so given time, the always start to separate.