The biggest risk of injury comes from overuse, particularly if you are getting older. Body weight exercises are very safe, and the biggest real risk of not doing them correctly is wasting time because you won't get the results you want.
The key to pushups is keeping your body straight as a board. If your body is sagging in the middle, it's likely because your abs are too weak. If you do some situps to help strengthen the abs, you should be able to get to the point of doing a proper pushup soon. Another approach is to do your pushups from your knees until you can do those with proper form.
Pullups are great, and even if you do kipping pullups or jumping up to get started, you will get stronger. Eventually, you will want to do them with strict form, but it takes time to be able to do that.
Doing body weight squats with bad form also won't put you at risk of injury. You just won't be getting the benefit from them that you desire. If you work on breaking from the hips, and going as deep as you can, you will get more benefit from them. Eventually you can work up to squat-jumps where you squat down, and jump up from that position as high as you can.
There are other body weight exercises that are quite useful such as:
- Chair dips. Not everyone has two rails near each other where they can do proper dips (shoulders going just below the elbows, but not too much more than that). But everyone has a chair. Put your hands on the edge of the chair, and put your legs out in front. Now lower yourself until your bottom hits the floor and back up.
- Reverse pushups. If you have a bar, or a solid ledge you can grab hold of, lay on your back underneath it, and pull your body up to the bar.
- Situps. An oldy but a goody. Ab strength is important.
- Planks. Put your weight on your elbows and lift your body so that you are straight as a board. Hold that as long as you can. Even better than situps--but harder.
- Back raises. Basically you will need something to brace your feet as your body hangs over the edge of a table or something like that. Keep your hands behind your head. Lower your body, keeping your lower back straight, and then raise it as high as you can.
There is a rhyme and a reason for the variety. The idea is that if you exercise one side of the body, you should balance it by exercising the other. For example, pushups work your arms, chest, and core. Things that balance it are reverse pushups and back raises.
The balance to planks would be back raises. The balance to pullups would be dips.
If you can't do it with strict form just yet, don't worry too much about it. Just try to get more and more strict with form as you can. That will help you get stronger and stronger.