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The DA score doesn't really mean anything. You shouldn't be focusing your time or energy on it. It's a fake number that SEO tools come up with as a best guess. Nothing wrong with using it as a guide but it doesn't really matter if it goes up or down. It does not correlate to any actual number that Google may have for your website. It's the tools best guess as to how Google might view your site. Focus on individual keywords, how they are ranking and what kind of traffic you are getting. Those are things you actually have control of and can see real data on.

I am assuming you checked the basics like whether or not the page was set to noindex (unlikely if it's indexed in search console) or blocked by robots.txt. Is the back page related to the same content that the rest of the site is about? PA/DA don't really mean anything, they are just best guesses as to how Google might see the page. It's really hard to know what is going on without way more information like the URL and what keywords you are trying to rank for.

You can operate two different businesses with the same address. That wasn't always the case but it's been fine for at least a few years. I have two companies at my location, one is a Web Design and Digital Marketing company and the other is a full service Print shop. Two distinct businesses using the same address. Where this might become a problem is if my Web Design company moved into a building where another Web Design company existed. Google would likely have a problem and think you are trying to get two listings for the same search result.

I would recommend having some differentation like the phone number if you are going to use the same address. Both companies should have their own phone line.

Why take the risk? Having your site load with the www doesn't hurt you. It's not as if the site will perform better without the www and if your site is configured properly your site will load no matter how you have that part of it set.

Google views www.domain.com as a different site from domain.com. You can mitgiate your risk by doing 301 redirects from your www site to your non www but there is always going to be some amount of risk and the potential to see a temporary dip while Google figures out what just happened. Not sure I see enough benefit in making the move to warrant changing it.

We worked with a content site years ago that focused mostly on superhero movie news and a lot of their traffic came from Google News. One day it just dried up and dropped by more than 90%. They brought us on to help and what we figured out was they were feeding Google their sponsored posts for months. The posts were made to look like news but were very clearly sponsorships pushing product so they got hit with a penalty. Not sure if that is what is going on with you but it's food for thought. As Tom mentioned, they are very volatile channels.

Google will view it as one site so you shouldn't have any issues from that perspective. The Google bot is just looking at pages and won't know/care that the underlying CMS that is running some pages is a different install than other pages. The downside is you now have two websites to maintain, two themes, two sets of files, etc... That may result in a bit of a headache in the future.

We worked with a content site years ago that focused mostly on superhero movie news and a lot of their traffic came from Google News. One day it just dried up and dropped by more than 90%. They brought us on to help and what we figured out was they were feeding Google their sponsored posts for months. The posts were made to look like news but were very clearly sponsorships pushing product so they got hit with a penalty. Not sure if that is what is going on with you but it's food for thought. As Tom mentioned, they are very volatile channels.

Why take the risk? Having your site load with the www doesn't hurt you. It's not as if the site will perform better without the www and if your site is configured properly your site will load no matter how you have that part of it set.

Google views www.domain.com as a different site from domain.com. You can mitgiate your risk by doing 301 redirects from your www site to your non www but there is always going to be some amount of risk and the potential to see a temporary dip while Google figures out what just happened. Not sure I see enough benefit in making the move to warrant changing it.

Google will view it as one site so you shouldn't have any issues from that perspective. The Google bot is just looking at pages and won't know/care that the underlying CMS that is running some pages is a different install than other pages. The downside is you now have two websites to maintain, two themes, two sets of files, etc... That may result in a bit of a headache in the future.