You shouldn't need to push the clutch pedal all the way to the floor unless something is wrong. Stock engagement should be pretty much exactly as you are describing. Not to mention that routinely pushing the clutch pedal all the way to floor as well as "stabbing" hard at the pedal are good ways to stress the seals on your clutch master cylinder. I learned the hard way, as I went through them every 20K miles until I made a change in my driving habits (and an adjustment to my RJM clutch pedal).
The Z's 6MT is noticeably clunky for sure, and the cold weather probably isn't helping. It might be perfectly normal. Has the transmission fluid been changed? Fresh fluid might help, and you should consider Redline's MT-85 fluid, which a lot of people say helps the 6MT shift smoother
