The most common reason for getting constant log in requests is that your browser Internet Cache and Cookies are waterlogged and need to be wiped clean. Its a long standing problem with IE, I dont know about other browsers. It also means you will have to manually log in to all your bookmarked sites the first time around so hopefully you know the p***words. For email I use the K9 anti spam program. Its free and works great after it learns what you want. Mine now reports a 98.6% catch rate and I dont even see them unless I want to open the K9 log.
I have seen that Ebay has a real problem with fake or scam auctions. While recently looking to purchase a motorcycle, I ran across atleast 5 fake auctions which I reported to Ebay. The fake seller will have a buy it now price but you can not click on a buy it now ****on to end the auction. He or she will try to talk you into sending a Western Union to an address usually in another country. They will send you fake emails from Ebay and Square Trade, Etc... trying to gain your trust in it being a genuine purchase. If you bite and send money you will be screwed for sure. I have also seen 2 fake car auctions with the same scam. Its getting hard to use Ebay and feel completely safe.
Those two emails were most likely honest and from Ebay. Su****ious activity usually refers to you or another Ebayer emailing the other with a question. The other email is also from Ebay explaining any new changes in their policy, you have your ebay account set up to notify you of new changes. I have the same thing coming to me.
From their website: Account protection tips: Be sure the Web site address you see above starts with https://signin.ebay.com/