Yes, you should list all your jobs in the 5 years prior to the date you are putting on your application, even part-time jobs. If you have a paper trail (i.e. tax documents like a W-2) for a job, you should DEFINITELY list it.
I did not have any part-time jobs, but I listed time periods when I was a part-time student living off my savings, since being a student also counts.
I don't see a reason why switching jobs often should count against your application, unless you failed to file taxes on that income. You filed and paid all your taxes, right?! (Or filed and gotten a rebate, or discovered you have no tax liability... but the important thing is that you be in good standing with the IRS. I was asked about this several times on my interview.)
I'm not a lawyer and this is not legal advice, just my opinion and personal experience.