Not every document that needs to be signed needs to be notarized — only certain types of paperwork require a notary's seal. While laws vary from state to state, they typically include real estate transactions, certain legal documents, many financial documents, and some forms related to healthcare.
A motion is an application to the court made by the prosecutor or defense attorney, requesting that the court make a decision on a certain issue before the trial begins. The motion can affect the trial, courtroom, defendants, evidence, or testimony. Only judges decide the outcome of motions.
Generally, the answer is no (at least not for an extension of time). The only motions that need to be notarized are verified motions, such as summary judgment.
Note that the document should be notarized like any other affidavit. If the court does not require an affidavit, then a memorandum can be written instead. It can take the same form, need not be notarized.