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Yes it is of course possible, and an average computer is more than enough for single-user computations (of course you can always eat all the computer ressource with a dummy loop, like in any language). But advices about how to achieve that depends on your system, so you have to be more precise if you want more informations about that.

When running standalone, there is no such thing as "project", but it should be easy to export the indivicual worksheets of sagemath cloud, an "projects" will basically corresponds to the organization of your directories. You should export them in the .ipynb format if you want to run them from the jupyter interface, of in the .sws format if you want to run them in the sagee-notebook format (which is going to be deprecated). In any case, do not export them in the .sagews format which is specific to the sagemath cloud interface.