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No, it's an issue with Joomla! not copying all necessary files on update. It's only been around since Joomla! 1.5.5 but nobody has managed to fix it (yet?). Try installing each update package twice in a row, without uninstalling the software before or in between, and you'll see that now it will show as being up to date.

That's actually a different issue and now things make more sense. PHP times out while Joomla! is installing / updating software. Sort of using a faster host there is nothing you can do about it. Frankly, if your host cannot handle installing Akeeba Backup you should probably consider moving to a better host anyway. To put things in perspective, on a normal host the update takes about 5 seconds including download time. If yours is timing out it's taking 2 to 10 times as much. You can infer what that means for your site's performance.

To come back to your question. Nothing has really changed during the last few months. In fact, the size and layout of the update package has been quite stable since the new features don't amount to all that much new code. The problem is that the which files are not copied (i.e. the exact point where the PHP times out) is semi-random and depends only on your host. If it happens in an inopportune moment it can get your site down. Frankly, you were just very, very lucky until now.

That makes no sense. Once Joomla! starts extracting the archive it's just Joomla's code. The only reasonable explanation would be a CPU or disk I/O cap which causes the server to kill the PHP process. Ask your host to check their configuration. What you are describing definitely makes no sense.