ok from the top - no video with green light is a very common fault for the ps3 slim, also a reflow probably wouldnt go wrong to attempt to fix it but im not convinced it is needing reflowed. i assume you tried video out with different cables (hdmi, component, composite) and i also assume you did a video reset? (power of console power on and hold for second beep).
as for the red light problem there seems to be a weakness in the design of the psu in slims as it is more susceptible to red light simply due to the psu being slightly out of calibration. (and yes you can recalibrate it but once its underway its a loosing battle). the psu could cause both problems so i would start by replacing it or possibly you could try the recalibrate trick (search youtube).
also if you decide to do the calibrate trick be careful those psus can kill - for the cost i would just consider replacing the psu and just hope that cures it.