First Publication as a PhD student
A Professor here.  


In general, yes, aim as high as possible.  In practice, I do an educated guess as to which journal to send it to after seeing the final product/paper.  If the student is capable and motivated, then I would work with the student to get the paper into the best journal.
As much time as possible.  Papers only get better over time.   There are always avenues to improve it, technically or presentation wise.  This is where your supervisor's experience is critical.  Only he/she will be able to push your paper to the required standard and also steer you away from pitfalls.
Many directions here.  You could beef up the paper with new results, incorporate more assumptions or proof the general case :)  Alternatively, you can work on a new problem or a related problem. Problems tend to reveal themselves as you dig deeper.  All my students start on a new/related problem after they submit a paper.  They might choose to go on a holiday first though.


All the best ...