Unfortunately this question is answered using a question. Will the content be different? if not then NO. If you are planning to offer your site with different content, for example multilingual (ccTLD) then yes search engines will geo target those sites to customers of that language. If the actual content is not language specific then there will be no difference and even may create issues for your site

NO, they are completely ignored, the TLD is a pointer to a site nothing else. bestproductforyou.com has nothing to gain compared to random.com and the second one is memorable for visitors.

The only benefit will be by making it easier to get the domain name you wanted and it wasnt available on your current gTLD or if you want a smarter/smaller domain, there is no other benefit but there is also no negative effect. Note that moving your site can bring huge drops in traffic if its not done propertly.