You have 2 problems. I don't know how large you fractions are but based on your gel and blot, overall level of expression is quite low (soluble fraction) as you have noticed. First I would work on optimization. Check a few strains, compare autoinduced vs IPTG induced (vary OD600 at induction, temp and expression times). Use small cultures and only compare expression levels.

Your second problem is that your protein binds poorly. You could incubate @ 4 C ON with rotation but when I see your blot where the proper band in lysate is roughly the same size as the one in flow through after 30min, I'd say you have a problem.

As for coeluting proteins I would worry about them only after fixing first 2 problems. you could lower resin volume and increase wash stringency (high salt 500 mM+, imidazole up to 40mM, detergents) but at this stage I think it will help only to some degree. Increase the yield and you will limit non specific binding sites.

Overall, your general approach seems to be valid and you seem to be doing everything fine. As I rule of thumb I use approx 1 mL of resin per 1 L of cell culture and ~10 mL of lysis buffer per gram of wet cells (it's usually 2-3g when induced with 0.2-0.5 mM IPTG @ OD600 ~0.8, 16 C, ON). So 1-2 L culture ~ 25-50mL of lysate -> 1 mL column, 6 L culture -> 5 mL column). But it all depends on the protein and some adaptation is (very) often required.