> Would a space crane be slave to the same equations?
No it would not, it has its own cross to bear, tensile strength.
If you start from the ground, then as the crane gets taller, the forces on its outer edges are the torque applied when a lateral force pushes against it (think wind). As with a stubborn nut, the longer the wrench handle the more force you can apply to the nut. If you think of the crane as your wrench handle, the taller it gets, the more force it will exert on its mounts when air moves past it. (this it true of sky scrapers as well).
Eventually you move past the point where there is anything that can stay vertical (steels rip and bend, carbon fiber breaks, Etc.) There is a working solution, a pyramid, but to build a pyramid that reached low earth orbit requires a base that is larger than the state of Kansas and will sink down to bedrock. There is an entertaining discussion of this from 2011 here: https://www.wired.com/2011/07/does-the-slope-of-a-pyramid-re... We know from the Himalayas you can stack up a lot of rock into a really really tall mountain, but its really really a lot. This might occur naturally on some planet, Olympus Mons on Mars is 16 miles high (about 85,000 ft or almost 26 km) that is 25% of the way up to space (assuming the 100km is your line for 'space'). Eventually you hit the limit which is that gravity prevents you from having a feature stick out too far from a sphere.
Another way to do this is to put a weight in geosynchronous orbit so it stays over the same spot all the time. If you start building a cable downward and adding mass further out you can keep the "effective" mass at the geosyncronous point. Eventually, (materials science not withstanding) the part you are lowering down to the planet reaches the surface and voila you have your jack and the beanstalk type beanstalk.
As fantastical as that sounds, it should actually work, if, and it is a big if, you can make a material that can withstand the tensile force of being both pulled up from the counter weight, and down by gravity. Once built you could attach a vehicle that would ride up and down the cable moving things from ground level into geosynchronous orbit, with only the energy needed to climb up the cable. These "space elevators" are not precluded by physics so they show up in science fiction stories.
But once you get away from building structures you are left with either chemical rockets or some how pushing against the Earth. Creative ideas there are balloons that take you part of the way up, and electrostatic systems that use an electric field to push against the charge on the Earth. These ideas start off far fetched and move right into crazysauce fairly quickly.