Your setup looks good. Should work well for your requirements. Yes, all you need is wires. I think you should stay with the Behringer mixer. But if you wanted to go a little cheaper on the mixer you definitely want to get something that has decent vocal effects.

May want to consider something like this. https://www.amazon.com/Pyle-Interface-Bluetooth-Connection-PMXU83BT/dp/B0719BC5FS/ref=pd_sbs_1?pd_rd_w=pYSSE&pf_rd_p=2419a049-62bf-452e-b0d0-ca5b7e35a7b4&pf_rd_r=DWQXHEK6WS142AQ3X1E8&pd_rd_r=7ca261ee-77d5-4371-9230-1163de0fc901&pd_rd_wg=eZwlr&pd_rd_i=B0719BC5FS&th=1