How to limit input current with LTC3625 and MPPT circuit?
The way to control DC to DC converters like this is through their feedback pin, in this case the PROG pin. (Usually linear sticks to 1.2V being the control voltage). If the voltage is too low then the regulator increases the length or time of switching. Too high and the converter decreases length or amount of switching. 

The key is changing the feedback loop, topologies like this could convert this regulator into a CC\CV supply:

Precision CC CV circuit or power supply

However, this may affect stability. I'm not a power engineer by far. I did use a linear tech DC DC buck in a digital feedback configuration with a digital PID to change the buck from CV to CC mode for a battery charger with and MPPT tracker, and that worked pretty well. 


  Also the simulator only seems to work with Voutset to 4.4v. If I make
  Vout=0, then it does not reach the output voltage.


Check the value of the PROG pin, it should be around the value listed in the datasheet, if not usually they kick into some other mode.