Natsu isn't that tough, as long as you remember not to 66K. Keep her out and block her damaging starters. 44A, 4A, A+B, 5B, 44B, and 11B are your moneymakers here. Backstepping still hurts her. Her 66K is pretty annoying in this matchup, but block it for free advantage/punish.
Even though Natsu can punish 66K, it's still useful in this match up. 66K/66K:BE stops PO transitions. You're right on minimizing 66K's use on neutral. I find using 4B as a substitute, for many matches, works well. 3K variations are also worth recognition.
Backdashing VS Natsu is pretty risky. Same reason why grabs are risk(ier) in this match up. Her 66B kills backdash, is a TC, is i17, and has a tech step in it as well. Also leads to PO rush downs (66B4).
Also, ShinJin, I have found that Astaroth's CE is incredibly useful in this match up. It punishes her 66B, 66K, PO K, and every move that's -15.
Mitsu? I want to say ALL of his matchups are 5/5 because of how his gameplan works. It's all 50/50s. You guess right, you win. You guess wrong a few times, you lose. We get more guesses than him, so it's not too hard to win here.
Superior 50/50's is a dominating factor in every one of his match ups. The thing that really irritates me is his frames. His frames have low -'s. Mitsurugi can press a lot of buttons in this match up. As for his 2KB:BE, you need to JG the "B" to get his most damaging (guaranteed) punisher. Unless you opt to use a Grab attempt, or use meter for 66K:BE.
Mitsu also has superior step killers at mid range, compared to Astaroth. Basically, his 1B(step), 22_88A, 4A, 6A, 66B, 6B, B9, B6, and practically any other move locks down Astaroth pretty heavily, and kill backdash/step. Notice these moves are the basics of basics with Mitsurugi. Mid range, close range, Mitsu hurts Astaroth. Stepping can be effective, until your eating 40dmg for it.