Does encrypting with MGF1/SHA-512/1024-bit seed equal to a 1024-bit key block cipher? Suppose: 


Alice uses SHA-512 with MGF1(as in PKCS#1 V2) and a 1024-bit random seed to generate a mask, XORs the mask with a message(M), gets a cipher text (CT), and sends the CT to her old friend Bob. Of course, curious Eve is listening...
Alice and Bob exchange the seed in a secure way (while Eve sleeps and does not know about it); they do not care about the speed of this algorithm.


Questions:


Is CT as secure as being encryted by a 1024-bit block cipher? 
Why?
This is not a block cipher.  It is a stream cipher: you are using SHA-512 to generate a one-time pad to encrypt the message.

The security—meaning the attacker's expected cost to break it—is at most $2^{512}/t$ if there are $t$ users of the system.

How the best possible security of this system relates to the best possible security of a 1024-bit block cipher depends on the block cipher.  The idealization of this system—a uniform random 1024-bit pad—has the same security as an ideal 1024-bit block cipher.  But a practical block cipher has a key, and it is typically smaller than $\log_2 2^{1024}! \approx 1024 \cdot 2^{1024}$ bits long.

In practical terms, you should focus more on everything else about your application's security.  For example, you didn't mention authentication, which suggests I can possibly destroy confidentiality by forging messages like EFAIL.  There is no point in pushing anything beyond a 256-bit security level, if there was any point going beyond a 128-bit security level in the first place as you get with AES-256-GCM or crypto_secretbox_xsalsa20poly1305.