In homomorphic encryption, the cloud performs operations on encrypted data, and gets only an encrypted result. He does not have any information on the decrypted result.
That's the difference between FHE and IO or multilinear maps. The second ones are almost all broken, mostly because of the attacks you mention. FHE, on the other hand, has semantic security.
The "cloud keyset" is another name for "public key". Usually, public key enables encryption but no decryption, here, the cloud key enables computations on ciphertexts, but not decryption.
Like in public key cryptography, you can give a RSA public key to an adversary, he can use it as much as he want, he will not be able to get any information on the private key (in any practical time). Most importantly, the cloud key is not secret, nor obfuscated: the cloud already knows each bits of it (side channel attacks are not relevant).
The "cloud key" is the public part of the key given to the cloud so that he can perform his operations (circuit, bootstrapping). There are strong security proofs that show that it is impossible to recover the secret key from the cloud key.
In the case of TFHE, if there was a polynomial algorithm that would recover even a single bit of the secret key given the cloud key, this algorithm could be used to break the LWE problem, and also worst case instances of lattice problems (which are much harder than factorization or discrete log for instance).
The best analogy is that Helib is a homomorphic GPU while TFHE is a homomorphic CPU. Elementary operations (addition, multiplication modulo p) with Helib are slower (especially bootstrapping), but are performed on a huge vector of data simultaneously. In the opposite, elementary operations with TFHE (binary gates) are extremely fast, but deal with a single bit.
In other word, if the application you are aiming at is suitable for running on a GPU, go for Helib, else if it would be faster on a CPU, use TFHE.
The current version of the library achieves 13ms per binary operation on a core-i7 instead of 50ms in the original paper.
The improvements are described in the new paper: https://eprint.iacr.org/2017/430. Hopefully, the packing techniques will be also present in the next version of TFHE.
That's the difference between FHE and IO or multilinear maps. The second ones are almost all broken, mostly because of the attacks you mention. FHE, on the other hand, has semantic security.