If you are not being payed to make the design of the wall, mug, cup etc. Just send the file but with the bear in a diferent layer. That bear is not part of the pattern.
Send it in vectors, but in my opinion you should not cut the vectors. Put the full submarine and icecream but inside a clipping mask.
If you are being payed, no, you should prepare the file to the specific product.
But you need the specs from the privider. There are cylindrical mugs, conical, spherical, big, tall, small, fat, thin...
A wall can be painted, printed, pasted up with printed vinyl stickers or with cut out stickers, airbrushed with a mask, etc.
One recomendation is that you prepare a little "manual of usage".
The minimum repetitions of this pattern to be used is one. You should not use half pattern.
Do not change the color of the pattern.
If you want the pattern not to finish abruptly, delete the half submarines but only on the repetitions on the borders.
For a wall of 2 mts tall, the recomended dimension of each tile is 50cm.
For a wall use a tile 4x tall.
For a Mug use a tile 1X tall.
For a door, or a vertical oriented wall, use 1x tile at least at the base.
For a mobil wallpaper, for desktop.
This example would be a Mug 1x tall to whatever is necesary at the base, it can be 2.5 or Pi.
This way you do not really design the aplications, but you do not leave at random decisions the aplication of the pattern.