For the Ü try Alt+154 and for the ü, Alt+129, it works for me.
Not really a difference, the "estar" variant just sounds with a "stronger shade" to me. For instance, "tengo vergüenza" sounds to me like a kid saying "I feel shy" ("...to talk to that adult", for instance), while "estoy avergonzado" sounds more like strongly apologizing, meaning "I'm really ashamed of what I did." Same with "tengo celos" (which sounds to me as almost apologizing for being jealous) and "estoy celoso" (to me, closer to "I'm jealous as hell"). On the opposite, "tengo miedo" and "estoy asustado" sound pretty much the same to me. But this can be my personal perception or the particular variant of Spanish in my area and it would be highly dependent on the context.