The space elevators problems include not only finding a good cable material, but also solving the problem of collision with satellites, so I'm not holding my breath.
Aerospike nozzles are spike nozzles where the spike is cut short and "replaced by air". They are shorter and usually lighter than equivalent bell-shaped nozzles, but they have bigger surface area in the critical section - most heat-loaded part of the engine - so cooling them is harder. For small engines the problem is cooling, and for big engines there is a problem of area where to put those engines (for large rockets length and weight of nozzle isn't a problem, but cross-section of the rocket, where the engines have to be installed, is), so aerospikes have different trade-off than bell-shaped nozzles.
Aerospike nozzles are spike nozzles where the spike is cut short and "replaced by air". They are shorter and usually lighter than equivalent bell-shaped nozzles, but they have bigger surface area in the critical section - most heat-loaded part of the engine - so cooling them is harder. For small engines the problem is cooling, and for big engines there is a problem of area where to put those engines (for large rockets length and weight of nozzle isn't a problem, but cross-section of the rocket, where the engines have to be installed, is), so aerospikes have different trade-off than bell-shaped nozzles.