Is RC5 a symmetric code?

Does RC5 use a secret key to encrypt the symmetric encryption key in the RC5 algorithm?

If not, is this key length really a 128 bits? Not really. For most symmetric algorithms, the key schedule is public; there's no security benefit to hiding it or using a secret key to generate it. (Though it is true that cryptographers usually use one time pad for security just as it's hard to imagine a more secure algorithm.)
Of course, you don't need to specify what you are talking about - it could be any algorithm for this step.

Is RC5 block or stream cipher?

If the latter, why does it support only 256 bit blocks?

If my understanding is wrong: What is RC5's cryptanalytic weakness? That is true about RC5, and many others. They all come from a time where computing power was much less. But as computing gets faster, cryptanalysts are able to get a better look at the cipher in action. In this case, the AES cipher is the basis of the cipher which is used by all. It can be described as an 8 byte key based cipher.

What is the difference between RC4 and RC5?

For those who do not know, they are symmetric cryptography algorithms.

I always want to know and understand how they differ from each other. Are they used for the same purposes or different?

What you've found is the difference between the two names. RC stands for Rivest-Shamir-Adleman, while RC5 is a product and not a protocol.

There are a couple of common uses for the two types. RC4 is used to create RC5 (or in other words, to create a wrapper that implements RC5.) RC5 can be used to create other cryptographic primitives (eg, CBC), and many (such as OpenSSL) use both to generate new key pair and encryption or decryption logic.

The Wikipedia entry for RC5 seems more explanatory than the RFC itself: For example, it is possible to generate public/private key pairs using this stream cipher. The public key is an X matrix which can be used to encrypt data; the private key is a secret matrix, X, of the same size which can be used to decrypt the data.

The name RC5 derives from the fact that the matrix X in this instance is 5x5, but RC5 is typically used in situations where the matrix is of any size.