The reason it disappears is because the image only has a graphic when the Even faces "Down", but when it turns any other direction, like when it moves, it will disappear until it moves Down again. The easy way to get this to work is to enable Direction Fix, but that is not quite the Correct way to do it. Fill in the rest of the graphics there with rotated grapics. Then you can use Direction also to provide 12 frames of animation (or 16 if you have $Filename.png) instead of just 3. You will most likely need to use Direction Fix to prevent Turning While Moving, but that will both fix some stuff and cause issues elsewhere, namely, you wont be able to turn, but that is also easily overcome.
Even if Direction Fix is turned ON, you can still turn an Event by directly accessing the Property. Instead of putting in an Event Command to Turn Left, use a Script instead: @direction = 4 which is what holds the direction, but doesnt check for for Direction Fix (a.k.a. Direction Lock).
There is another property called @pattern. The @pattern controls which column will be displayed. So @pattern = 0 will display the stuff in the left most column. @pattern = 1 will display the graphic in the next column. @pattern = 2 will display the right column, unless you use $filename. @pattern = 3 will only work with $filename and then will display the right column for XP style sprites.
There are two ways to do this: either "Change Graphic..." and select which frame of animation you want, or use @direction and @pattern.
Example 1:
Change Graphic (Row and Column)
Wait 3 Frames
Change Graphic (Row and Column)
Wait 3 Frames
Example 2:
Script: @pattern = 0
Wait 3 Frames
Script: @pattern = 1
Wait 3 Frames
Script: @pattern = 2
Wait 3 Frames
Script: @direction = 4
Script: @pattern = 0
Wait 3 Frames
Script: @pattern = 1
Wait 3 Frames
Repeat until a full cycle is completed, but it gives an example of how to fully control your animations, even if Direction Fix is ON or Move Animation is OFF.
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In regards to Move / Animation Speed, I wrote a script for XP that could probably also be done for VX Ace, which I cant do since I dont own VX Ace. Basically, it allows you to seperate Move Speed from Animation Speed so you can have an NPC that steps very quickly even if their Move Speed is very low, or vice versa. This is usually great for Children or small sprites as they would have to step more quickly than taller characters to move the same speed.
Heretic's Animation Speed [XP]
http://forum.chaos-project.com/index.php/topic,13777.0.html
Again, this probably wont help VX Ace users, but the concept is simple and can be easily ported over to VX Ace. What you'd do is put a Comment in that says anime_speed=6 and that will override Move Speed for anything related to how quickly they appear to step.