You've asked a few questions here...
Instead of deleting the memory what kinds of benefits would we gain based on the reuse of the objects?
That depends entirely on your application. Even supposing I knew what the application is, you've left another detail unspecified -- what is the strategy behind your re-use? But even knowing that, it's very hard to predict or answer generically. Try some things and measure them.
As a rule of thumb I like to minimize the most gratuitous of allocations. This is mostly premature optimization, though. It'd only make a difference over thousands of calls.
What is the process of new?
Entirely implementation dependent. But the general strategy that allocators use is to have a free list, that is, a list of blocks which have been freed in the process. When the free list is empty or contains insufficient contiguous free space, it must ask the kernel for the memory, which it can only give out in blocks of a constant page size. (4096 on x86.) An allocator also has to decide when to chop up, pad, or coalesce blocks. Multi-threading can also put pressure on allocators because they must synchronize their free lists.
Generally it's a pretty expensive operation. Maybe not so much relative to what else you're doing. But it ain't cheap.
Does a context switch occur?
Entirely possible. It's also possible that it won't. Your OS is free to do a context switch any time it gets an interrupt or a syscall, so uh... That can happen at a lot of times; I don't see any special relationship between this and your allocator.
New memory is allocated, who is doing the allocation? OS ?
It might come from a free list, in which case there is no system call involved, hence no help from the OS. But it might come from the OS if the free list can't satisfy the request. Also, even if it comes from the free list, your kernel might have paged out that data, so you could get a page fault on access and the kernel's allocator would kick in. So I guess it'd be a mixed bag. Of course, you can have a conforming implementation that does all kinds of crazy things.