I am a student and I have small knowledge on C++, which I try to expand. This is more of a philosophical question.. I am not trying to implement something.
Since
Since dealing with Exceptions is heavier compared to a simple if(t), why isn't the normal new T() not considered less good practice, considering we will have to use try-catch() to check if a simple allocation succeeded (and if we don't, just watch the program die)?? What are the benefits (if any) of the normal new allocation compared to using a nothrow new? Exception's overhead in that case is insignificant ?
The penalty for using exceptions is indeed very heavy, but (in a decently tuned implementation) the penalty is only paid when an exception is thrown - so the mainline case stays very fast, and there is unlikely to be any measurable performance between the two in your example.
The advantage of exceptions is that your code is simpler: if allocating several objects you don't have to do "allocate A; if (A) { allocate B; if (B) etc...". The cleanup and termination - in both the exception and mainline case - is best handled automatically by RAII (whereas if you're checking manually you will also have to free manually, which makes it all too easy to leak memory).
Also, Assume that an allocation fails (eg. no memory exists in the system). Is there anything the program can do in that situation, or just fail gracefully. There is no way to find free memory on the heap, when all is reserved, is there?
There are many things that it can do, and the best thing to do will depend on the program being written. Failing and exiting (gracefully or otherwise) is certainly one option. Another is to reserve sufficient memory in advance, so that the program can carry on with its functions (perhaps with reduced functionality or performance). It may be able to free up some of its own memory (e.g. if it maintains caches that can be rebuilt when needed). Or (in the case of a server process), the server may refuse to process the current request (or refuse to accept new connections), but stay running so that clients don't drop their connections, and things can start working again once memory returns. Or in the case of an interactive/GUI application, it might display an error to the user and carry on (allowing them to fix the memory problem and try again - or at least save their work!).
Incase an allocation fails, and an std::bad_alloc is thrown, how can we assume that since there is not enough memory to allocate an object (Eg. a new int), there will be enough memory to store an exception ??
No, usually the standard libraries will ensure, usually by allocating a small amount of memory in advance, that there will be enough memory for an exception to be raised in the event that memory is exhausted.