For most departments, the only automatic disqualifiers are felonies. All other offenses are investigated and handled on a case-by-case basis. Check with the specific department to be certain.

Each discipline is different. For the fire service, there are several certifications that are typically needed before applying, which will take longer. Police and fire departments also typically have more evaluations, interviews, tests for candidates to go through. These processes may take several months. Applying for the jobs that do not have the extra requirements will also vary on the agency. Applying for one of these should be a relatively quick process, covering several weeks.

Just like any job search, you should apply for all departments you want to. Your first choice might not work out, so you want to have options. 

Try volunteering with non-profit organizations in your intended field. Reach out to agencies/departments about internships, mentoring opportunities, volunteering, etc. See volunteer and experience information on the Training Resources page.

In most situations, public safety careers are performed at a designated work site and remote work is rare. However, some fields do allow remote work. It will depend on the specific role you wish to pursue and the agency's policies.