Should designers work with shapes (shoe lasts) strictly?
If you have a last shape this is ideal, but not necessary for just sketch work. As you move into a more finished design/rendering it can help a lot. If you are using a last, besure to allow for material thickness.
Or can also submit proposals for designs in the form of a certain style?
You can use a certain style if you like. It is good that you started with an existing shoe shape, but you will want to stylize your drawings a bit to make them visually more appealing. A lot of designers will tilt the heel forward and make the toe sleeker. Even the adizero sketch does this