The existing answer is very good. I'll provide some technical details, for those that like such things.
\Windows\System32) is the program that is run when you call for Task Manager. Inspecting it with a hex editor, I discovered that its manifest sets
highestAvailable. This means that if you are running as a local admin, Task Manager will require you to elevate. You can easily demonstrate this by setting UAC to the highest level and pressing Ctrl+Shift+Esc to launch Task Manager, noting that it produces an elevation prompt. If UAC is not at the highest level, Task Manager can silently elevate because it's an integral Windows component. In short, yes, Task Manager runs as admin by default when possible.
highestAvailable (as opposed to
requireAdministrator) allows non-admins to run the program without being asked to elevate, but they of course won't be able to do anything administrative from it.
A quick-and-dirty way to see if a program is running elevated is to enable the UAC virtualization column in the Details tab of Task Manager. If and only if a process's entry in that column is Not allowed, then it is elevated. (Administrative processes cannot be compatibility-redirected.) You could also check whether it has the powerful privileges (e.g.
SeSecurityPrivilege) using Sysinternals Process Explorer.
In response to your question about still being denied access to some processes, Windows has a concept of protected processes that absolutely cannot be touched from user mode, not even by processes running as
SYSTEM; the protection is enforced by the kernel. Only essential system services get this kind of guarding. One such process is