If the JPG background was EVER saved with compression then the compression remnants will show up even if you now save the image with zero compression.
Do not turn the banner into a JPG for printing. Save it as a PDF. As a PDF the vector parts of the banner will stay vectored and print the best they can. Likewise for the JPG parts of the banner.
Depends.
First, when a JPG is saved with or without compression it becomes a new JPG, there are no 'remnants', it is what it is. Unless the compression was somewhat severe you'll never notice the difference unless you're printing a copy of the Mona Lisa and there will be a side by side microscopic comparison. Every time you save a JPG you get a new JPG, still it's always best to do so with no compression.
Next, a PDF is supposed to be merely a container for disparate objects but with non-bitmap objects there can be an annoying color shift. If you turn the entire thing into a JPG, no compression, then you'll know pretty much exactly what you're going to get. Unless whomever is doing the printing does something silly like converting it to CMYK before printing it.
Whether or not to convert an image to a bitmap depends for the most part on whether or not it's going from your screen directly to the printer or it's going to a file to be sent to the printer. If the former, just print it. If the latter, convert the entire thing to an RGB bitmap. That's been the rule around this shop for years and it's worked out just fine. No complaints to date and I'm about as critical a person as you're ever likely to come across. Remember that the client usually has never seen the image outside of looking at a monitor or some low resolution proofs and has no idea of what it's supposed to look like. It's self-defining.