Can I make a brick bbq not built in?

They have blocks that interconnect, may be something you wish to look at as opposed to brick. No mortar needed then.

I built mine dry - literally stacking the bricks in a bond pattern, so that the whole thing could be moved and used a paving flag to make the fire bed.

To build as pictured, with mortar, it needs to sit on a concrete slab. Otherwise it will sink and crack up

Good for you.  It is time that you actually do some physical work so that you will appreciate all that you have.  If you look at that photo of the nice brick fireplace, you will see that it is brick and mortar.  It has to sit on a concrete foundation that is strong enough to support the weight without the motar cracking.  That requires digging a hole about a foot deep and a foot wide where the bricks will be set, and across the bottom where the fire goes.  So let's say a hole that is 3 feet x 3 feet x 1 deep.  All nice and shaped and all the loose dirt out of it.  Go dig that hole and see if you want to change your mind. If you do, you can fill the hole back up.   If not, tell you dad you want to proceed on with the forming the concrete and buying bags of concrete mix to fill the hole.  You get the bags back to your hole. Then put the mix in the hole, add water and mix it up.  Build it up to a flat foundation for the brick work. At that point, you should question yourself about making unnecessary work for a man.  If you still haven't figured it out, go buy the bricks and motor and build the fireplace.  Your dad will show you how to cut bricks.