Is it safe to reuse a symmetric key when using asymmetric encryption?
The use of the AES key many times is not a problem. However, there is a fundamental flaw with your solution. The server has no way of validating that it received the client's authentic public key. In particular, a man-in-the-middle can capture the client's public key, can forward its own public key to the server, and can then decrypt all traffic sent by each party (since it will be able to decrypt the AES key). Also, the client cannot know that it is talking to the authentic server at any stage.

Apart from that, you lose a lot of properties that people like to have in such protocols, like forward security.

In general, it's not a good idea to make up your own protocols (no matter how simple). My recommendation is to just use TLS v1.2 with a forward-secure cipher suite. The best to use are TLS_ECDHE_ECDSA_WITH_AES_128_GCM_SHA256, or TLS_ECDHE_RSA_WITH_AES_128_GCM_SHA256. However, you still have to solve the problem of somehow enabling the parties to identify each other, and to be sure that the correct public key is used. This is handled by using certificates.