Urgent - Help creating an SVG, and adding colour to AI illustration


I have a client who has asked me to create some digital illustrations in AI. I need to save them as SVG files. However, my clean-looking image suddenly shows tiny lines and all the ugly ‘backstory’ that went into creating it once I save it. Help! Any idea why this happens and how to avoid it?

Also, is there an easy way to add colour to my line work (created in AI)? Can’t use PS as the whole thing has to be a vector. I use the bucket tool but it doesn’t work for areas with gaps, and it also adds the colour to the same layer as the line work - I can’t do it on its own layer. It’s also too time-consuming to make coloured shapes with the pen tool to place behind the lines.

Any ideas welcome! Thank you!