A couple of things:
Although I can do both of these moves, my husband can't. He's a lot stronger than I am, so don't feel like a lack of strength is the problem. They're both moves that require a lot of flexibility and many people (male and female) don't have it naturally and it needs to be built up. You are correct on that.
I am flexible enough for an elbowstand press up, but don't have the strength in my core to do it yet. So I still kick up into elbowstands and handstands against the wall. My husband trains handstands both against the wall and in the middle of the room, and he kicks up into them like a gymnast would. His hamstrings are way too tight for a press up, though he may have the strength in his core.
Kicking up against the wall is OK as long as it is controlled. If you go SMACK into the wall then you are pushing too hard and need to go more gently. It's better to push, miss, and come back down with control than it is to go up with too much force. Those small attempts will help build strength and control. If you are falling/flailing then you are going too hard. If you think a video would be helpful, I could take one.
For bridge, you can work on some other variations to build flexibility in your spine without having to push yourself up all the way.
One is supported bridge:
With hands: http://pad2.whstatic.com/images/thumb/5/5f/Assume-the-Supported-Bridge-Pose-in-Yoga-Step-4.jpg/550px-Assume-the-Supported-Bridge-Pose-in-Yoga-Step-4.jpg
With a block:
http://res.mindbodygreen.com/img/ftr/Supported-Bridge-Yoga-Poses.jpg
The important things are to keep your knees close together (don't splay your legs to the sides) and try to take the move out of your back and glutes and into your quads. This is much easier to learn while doing the lower bridge position, and then apply that once you're strong enough for the full backbend position.
The full wheel pose (what many people call bridge) is an advanced move in yoga, requiring a lot of flexibility not just in the back, but also in the shoulders and upper back in order to get that "upside down push up" position. I have actually injured my back from doing that move with improper body mechanics. So definitely work on supported bridge or regular bridge first, it will help you gain back flexibility with less risk.
In the modified positions above, the weight is on your shoulders and upper back, not your head. If you're up on your head, that is incorrect. Don't do crazy things just to feel more flexible in your back… I used to be guilty of that kind of thinking myself and slipped a vertebra for the trouble. By no means should you be resting all of your weight on your head. I sometimes use my head as an extra "hand" when lowering myself down from wheel pose, but that's just a quick thing. Putting your body weight on your head can lead to a serious neck injury. (PSA over)
Good luck and welcome back!