Suspension does not affect tunes in any way. Tunes are made to the ECU, ( I think that stands for Engine Control Unit), so it only affects the engine, many tunes also affect shifting, which would obviously be your transmission.
Yes, upgrades to your suspension would be good now. I'd do that before a tune, as these cars have wheel spin like crazy stock. Proper suspension will make the car feel much different, and better. I think the stock suspension is too weak for the stock power level. Also, tires too. The Pirelli tires can't hook worth a d*mn, for all seasons, I'd recommend Michelin AS3 tires. Summer tires, I've heard good things about the Mickey Thompson Streets.
2013 GT Premium, Red Candy Metallic, Recaro seats, Electronic package, Strut tower brace, Sway bars, SHR flush mounted window louvers, RTR street-spec axleback exhaust, Ford Racing Boss 302 Side Exhaust, RTR wheels, 50/35% Window tint, Steeda black shorty antenna, and Ford OE Matte Hood Vents.