Is it sufficient to wash the penis with water only after urination? And what is the ruling on the discharge that appears on the outside of the urethral opening after repeated squeezing, and the fear of it exiting later with movement?

If you are afflicted with waswasah (whispers/obsessive thoughts), its cure is to turn away from it. When relieving yourself, cleanse yourself immediately without squeezing the penis, because the more you squeeze it, the more drops will emerge. Ibn Uthaymeen, quoting Shaykh al-Islam, said: "The penis is like an udder; if you milk it, it will yield, and if you leave it, it will settle." The best way is to perform istinja' (cleansing after relieving oneself) without bothering with the waswasah, and to build upon the fundamental principle, which is the persistence of purity.