I was wondering if I add a report feature and a way to take the content down if you are the copyright owner will companies use this or will they go straight to my web host?
Depends on how visible the "Report" feature is and also how large/aggressive the copyright holder is. In my personal experience, a well-placed and easily visible reporting button will get used most of the time because it's easier on all involved. Filing a true DMCA takedown is brutally effective but has some negative blowback on the filing entity as Chilling Effects will list them and that gets [insert community here] upset and potentially activated. If you agree to quietly remove the infringing content based off an email then none of that comes into play so both parties benefit (them more than you, of course).
That being said, a sufficient large copyright holder will go straight to the ISP because they will have that contact info on their speed dial and the form letters ready to go. I would urge you to read up on ways to protect yourself:
User Upload of Images & Copyright Issues
Responsibility in case of copyright infringing user-submitted content
Also, is my web host likely to suspend my services if I get a lot of reports but I deal with them in a timely manner?
Every host has a different tolerance level for this, but generally speaking, yes. If you become a problem for the ISP, they will glady cut you loose to cut their potential losses.