No. When you're running dist-upgrade it will only upgrade your system according to the current software repository. That being said it will not upgrade 13.10 to 14.04. The do the full distribution upgrade, you will need to change your software repositories to 14.04.

If you run "sudo apt-get linux-generic", it will only upgrade the kernel. If you run "sudo apt-get upgrade" however it will upgrade all packages that have an update & it may include the kernel as well.

Technically speaking, yes, the kernel (Linux) is the operating system. That is why the upgrade option for apt-get is dist-upgrade which stands for "distribution upgrade". Kernel (Linux) + all applications make up the distribution. If you upgrade from Linux kernel version 3.12.x to 3.13.x, you have technically upgraded the "operating system", but you haven't upgraded your "distribution" which might still be Ubuntu 13.10 (and not Ubuntu 14.04).