Question 1: It is possible that the company had paid to have a Web submittal company submit the name and URL for the site to search engines (Google, Yahoo!, etc.). It could very well be that the submitting company may still be submitting the site to search engines monthly if the firm pre-paid for a Yearly subscription or the like. It could simply be that the keywords that the old company site is using is “trumping” your keywords. You can find out the keywords the old company used by viewing the website and selecting “View”, and “View Source”. Look for a Keyword category and see the list. Also, if the domain name is still active (www.yourcompany.com) for the old name, and you have access to that record, you can deactivate the URL and that site will eventually drop off the search engines.
Question 2: Both topics are related. Keywords help you get listed on search engines, however if there are numerous companies that are in the same line of work in the same area (Automotive Repair in Dallas, GA), you won’t necessarily be at the top of the list. You could pay a company to continually submit your company name to the top of lists, but from what I have experienced, it really doesn’t help a lot, and it can be very expensive.
Hope this helps.