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It should be easy to install a 404 page that records the agent and if you notice googlebot is still hitting these pages, no harm in 301ing them to let google know that we have moved and it doesn't need to bother about this urls anymore.

That's not too difficult to handle. Here is how I would go about it.

I would first fix the existing website 404s, yes you have some as I checked your website.

next, I will set up the 301s from old website urls to new urls and also install a 404 monitor plugin.

also I woulld suggest that without deleting the copy that we have in old subdmain, if you can 301 everything there to www instead of old.domain.com, that will help.

Iv recently (about 20 days ago) updated my website and added scheme for local business but yet to see any effects. In fact, the rankings have dropped for what I entered as business description.

Yes switching theme means switching the HTML that the crawlers see on the page but "affect" does not necessarily mean negative effect. I believe all the good designers these days do semantic HTML which is crawled better and gets better results in terms of SEO.

So in answer to your question, everything else (content and URLs) remaining the same, the switch in theme can affect and also improve your rankings if done right.

Yes switching theme means switching the HTML that the crawlers see on the page but "affect" does not necessarily mean negative effect. I believe all the good designers these days do semantic HTML which is crawled better and gets better results in terms of SEO.

So in answer to your question, everything else (content and URLs) remaining the same, the switch in theme can affect and also improve your rankings if done right.