I had the exact same problem. I've been doing yoga for a long time, but got stuck on a casual level. I became more limber and my posture improved, but I didn't get any stronger. I've now done bodyweight exercises for the past three months parallel to yoga and my form has improved a lot. I'm now close to doing poses like half moon and scorpion which were completely out of reach for me before. I just have to work a bit more on my balance.
For things like shoulder and head stands you need a strong core. There are workouts in the app for learning these poses. As for crane, focus on finding your balance point before lifting your feet off the ground. Work through the pose gradually, lifting your feet more and more each time. Once you find your balance it's actually not that difficult. First time I tried it, during a yoga class at the gym, I fell over straight into a wall and hit my head, lol.
Just stretch, stretch and stretch!
We all know it's not as easy as that, but keep motivated and you'll get there. Start easy! Your body will eventually adapt and your joints will become more flexible.
A couple weeks ago I never thought I'd be able to do more than 2 pull ups, and three weeks later I can now complete 5 with good form! Best of luck