Quick and dirty answer - climbing with constant IAS means TAS is increasing, ie you are accelerating the aircraft, hence lower climb rate. Climbing with constant mach means you are decelerating in TAS terms. The performance guys decide what IAS and mach to fly and there is a cross over altitude where, on climb, you attain climb mach and fly that while the opposite happens on descent - you fly planned mach until you reach planned IAS then fly that.
As for icing, wing TAI is rare in my brand of jet above about A100. Engine anti ice should be treated as just that - ANTI ice, so get it on early when icing conditions exist. Bent fan blades are so expensive! Landing lights give you an idea of vis and hence potential icing. Individual engine types vary greatly in their sensitivity to icing - the old P&W JT9-3 wouldn't carry enough ice to chill your Scotch while the RR RB211 seems much more tolerant.
PS I fly big Boeings but I think the basics are very similar.