Agar is not a good choice for pudding because it makes a brittle gel and it won't melt in your mouth at body temperature. What you want for pudding is a starch based thickener. What we call pudding in the US at least is typically thickened with cornstarch. Modified starches like Ultra-Tex 3 can also work well. Are you thinking of something more along the lines of panna cotta, which does normally have gelatin in it? If so, you might try carageenan instead of agar. Here's a recipe that looks promising: http://irishherault.wordpress.com/2009/12/24/vegetarian-panna-cotta/
Fondant is nearly solid -- it's rolled out, and placed onto the cake (with a buttercream underneath as an adhesive). The advantage is that it goes on really quickly and smoothly over large items. It can also be cut into shapes with rather clean edges (modeling chocolate gives sharper edges, but it's not flexible enough to cover a round cake). You often see it on professional non-traditional cakes (more novelties, when they need a solid color or something that's strangely shaped.
Fondant can also be used to give a solid background can then be piped into (with buttercream icing, generally) or painted. As it's nearly solid, it can also be worked like you would clay to make decorations.
Royal icing dries hard as it has no fat in it. It's generally only used on cakes as a drizzle where you're not attempting to cover the entire cake. You'll often see it used on bundt cakes, but you more frequently see it used for cookies when you're doing a flood-fill.
Buttercream icings (commonly called 'frosting' in the US) is likely the most standard for non-professional stacked cakes. It's spread on, so it takes a little more work to get a clean smooth surface. (the trick is to wait 'til the surface has hardened, put down some waxed paper, then smooth out all of the spatula marks ... you need to wait 'til the waxed paper doesn't stick, but not so long that it hasn't set up completely). You can also use buttercream in pastry bags to create flowers, ropes, basketweave and other decorations.
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And then there's other frostings/icings, too, but they're not as general purpose. They generally go with specific types of cakes -- caramel and other cooked frostings (eg, used in 'German Chocolate Cake'), cream cheese frosting (for carrot and other spice cakes), whipped cream (used in black forest cake & strawberry shortcake)
Best Answer
Yes, there are several vegetarian fondant recipes using agar. Most tend to substitute it for gelatin in equal amounts.
Keep in mind two things:
Agar has a much higher melting point and sets firmer than gelatin, so the resulting icing will be less "melty".
The hydration process is completely different from gelatin. It doesn't bloom; it needs to be dispersed in cold/warm water and then heated all the way up to 90° C with constant agitation.
I'd recommend that you include the glycerin in any agar-based fondant recipes - they'll make it a lot easier to work with. Some gelatin recipes leave it out but IME you need it more for agar.