Did I turn my frosting into mayonnaise?
It might have the consistency of mayonnaise but so do many sauces, sweet or savory. If it tastes good to you, it's fine so use it.

What I do when I want a tart citrus taste (for many things, not just icing) is add citric acid crystals. I consider citric acid to be essential in a well-stocked kitchen. If you'd like it to taste like lemon or orange, add some finely grated peel of which ever you prefer. I find a slightly tangy orange flavored icing goes wonderful with chocolate cake. 

I can't say how much citric acid to add as I rarely use a recipe. When you're adding butter and whipping the icing, I think that's the best time to add a little citric acid (and peel, if you wish). Start off with just 1/8 teaspoon and after incorporating it well, take a taste. Start adding it well before you've whipped in all the butter. Keep adding 1/8 teaspoonful at each time (or less when you think it's almost right). Keep track of how much you've added so you can write it down for next time. Use a little less if you include orange zest as oranges aren't meant to be as tangy as lemons. Personally, I like tangy taste to many foods.

I use salted butter in all my baking too and simply don't add any extra salt. I like salt. I've asked friends if they found it a little salty and no one said they did.