I’m assuming you’re talking about the graham frosting. If so, then your butter and cream cheese are simply to warm and are in a melted state from all of that whipping. Pop it into the refrigerator until it is good and cold. Then give it a short whip and you’ll have the right frosting consistency.

Meringue is notoriously unstable and is known to weep (collect excess moisture). But this recipe uses a Swiss meringue, which is somewhat more stable. While it will slowly deteriorate, it should last 2 days or so if properly store in the refrigerator. A cake dome or light wrap of plastic will help as well.

I substitute vanilla extract and vanilla bean paste interchangeably. Ratio 1:1.

It’s a French star tip, like an Ateco 869.

Opps! The vanilla should be added at the end. I’ve updated the recipe accordingly. Thank you for pointing it out.
 

As for your meringue – meringue can be so finicky! Make sure you’re using a glass or metal bowl, start with room temperature egg whites and make sure your mixture is free of any traces of oil, fat or egg yoke. Even the smallest drop of fat will make it harder to create a stable meringue. Some bakers go to the extreme of wiping down the bowl and whisk with lemon juice before starting. As a last resort, you could try adding a stabilizer like cream of tarter or an acid. To do so, you’d add cream of tartar (about 1/8 teaspoon per egg white), or lemon juice or white vinegar (about 1/2 teaspoon per egg white) before beating. Hope this helps you troubleshoot!

It’s a French star tip, like an Ateco 869.

Sure, if you’d prefer butter. But you might be surprised by oil! Here’s an experiment we did comparing butter vs oil vs shortening. https://thecakeblog.com/2012/05/is-butter-better.html