Why does fan increase slowly its speed at boot and system is so slow?
Go to iFixit and look up the guide for your Air. You will need to do two things. First is disconnect the battery, second is remove/clean the CPU heatsink before reapplying thermal paste and refitting it. When you've done that you can reconnect the battery and see what happens.

SMC resets using key combos don't always work - disconnecting the battery does. It might seem drastic but removing the power source is what's needed. If you've an SMC issue (which sounds very likely) this should clear it and cleaning up the CPU cooler will give you something useful to do while any residual voltage is discharging.