Hall of Monuments

Let's say I fill up the mini pet display in the Hall of Monuments to its maximum. I find a rarer mini and I want to replace one of the displayed minis with it. Is this possible? How does it work?

There is no limit to how many minis you can have in the HoM, only a limit to how many are displayed. You can re-arrange your minis to display different ones, but it can be a slow and annoying process to get them all where you want them (or close).

GGs

Can you take minis/armor/weapons/titles/companions down once they've been put on display? Or can they only be replaced with another of its kind?

You can't take them down, you can only rotate those that have been added. Say for example you can display 20 minis and you add another so there are 21 on the list, you then rotate the display of 20 to show the ones you want.

In that case, what's going to carry over into Guild Wars 2, the ones on display or the ones on the "list"?

Unknown, but its been suggested that rather then carry over, it unlocks some other bonus for you in GW2.

The whole list.