Let's face it, Trelawney is a crummy teacher. But to date she's had two real prophecies. This, however infrequent, is a powerful tool. So Dumbledore takes her in under the pretence of teaching in order to protect her from the Death Eaters. Voldemort wouldn't have been too interested in her as he doesn't know about the second prophecy and he was only concerned with the one in the Department of mysteries. Now that prophecy is gone so he may want to go after Trelawney again in the hope of extracting it from her. It may feature in book 6.
Umbridge on the other hand displayed nothing but the utmost devotion to the MoM aka Fudge. My impression was that she truly dispised shoddy and fraudulent teaching and so went after Trelawney on that basis, to improve the image of the Ministry. Being the only Wizarding school in Britain (or so it seems
) parents would care a great deal about the standard of the school. Dumbledore stepped in and saved Trelawney from leaving, because he knew her real value and the dangers of her being captured. It also displayed that he wasn't yet under the control of the Ministry.
Umbridge's vendetta against Hagrid is very different she has a known prejudice against non-humans and half breeds. Also Hagrid was a well known supporter of Dumbledore and she knew about his mission to the giants. She could have got rid of him sooner, but then he wouldn't be under her watchful gaze.
As for open war, I wonder if we're going to see a full scale assault on Hogwarts
. Harry's death will still be the number one objective for Voldemort. He can't gain a true victory without it. Will there be guards in and around Hogwarts again as was in PoA? Wizard patrols perhaps. It will be exciting to see a fully mobilised Ministry in action. The real world equivalent of this situation is a military seperatist group taking on the government.