I’ve been riding mostly fixed gear for, oh, maybe ten years now. I use bmx pedals with straps.
You really ought to practice getting in and out of them until it becomes second nature. Having problems getting clipped in or getting your foot through the strap means you won’t have a rear brake (not ideal). Having problems getting out means you end up falling when stopping (as you noticed). That could be potentially quite dangerous. Do NOT use any kind of foot fastening system that you need to use your hands to get strapped into or out of! That might be fine on an indoor track, but is a phenomenally bad idea for road riding.
If you find your foot getting stuck when pulling it out of the straps, it might be that your straps are too tight or you’re wearing the wrong kind of shoe. When using straps, the best kind of shoe is—in my opinion—canvas shoes with relatively thin soles and without too much pattern, such as Converse. Modern running shoes tend to have quite bulky, grippy soles which means you can get stuck. Classic leather shoes tend to be wide in front and narrow in the middle which means it can be a bit of a struggle to get into the strap. Boots are just a bad idea for cycling, full stop.
Try practicing when leaning against something you can hold onto, like a lamp post or a railing; or on ground where falling down doesn’t hurt so much, such as grass.
This is simple: use your front brake. I know the cool kids on their fixies ride brakeless, but honestly, it’s just a bit stupid (or at least reckless). I don’t care how good anyone think they are at stopping their brakeless fixie: the fact of the matter is that the front wheel has much better stopping power than the rear. On a road bikes with front and rear brakes, you still mostly use the front one. Braking at speed adds significant downward force on the front wheel. More downward force means greater braking potential before skidding. On a grippy surfacy (dry road) with decent tyres, your front wheel simply will not skid. Even on my gravel bike with very grippy 35 mm tyres, it doesn’t take a lot of braking power to lock up the rear wheel. If you brake hard enough to lock the front wheel, however, you don’t skid—you go over the handlebars. While not ideal, it’s leagues better than going full tilt into a little old lady crossing the road because your rear-brake-only bike just skids.
Resisting the forward motion of the drivetrain is great for keeping your balance and controlling your movements at low speeds. When weaving between pedestrians or between lines of cars stuck in traffic, a fixie is terrific. When stopping at a red light, a fixie makes it easier to stay on your pedals (track standing), which means you get a quicker start when the light goes green. But for stopping at high speed, your front brake is your best friend.
Beware of pedal strike
One thing you need to get a feeling for is cornering. Because the pedals keep spinning even as you lean into the curve, you risk striking the road with your pedal. The faster you go and the tighter the turn, the more you lean over. Striking the ground that way can easily knock you off your balance. Also be wary of other potential obstacles. A few years ago, I hit the side of the pavement with my pedal. The force was enough to lift the rear wheel maybe half a metre off the ground. I rolled maybe 10–15 metres on my front wheel, at around 45 km/h, surrounded by taxis. It was terrifying. Don’t do that.