Can a 737 be on final at 160 knots, and also touchdown at 160 knots?
You fly final approach at the "reference speed" (Vref) for the flap setting which varies with weight.  You will normally set up the reference speed for full flap on the speed bug on the speed tape on the primary flight displays, and on final while the autopilot flies the glide slope you will modulate thrust to hold the approach speed (or let the autothrottle do it if equipped).

For a given landing weight, the reference speed for full flap is worked out from the "speed book" or "speed cards" which the crew usually has handy on the center console.  The airplane on short final with landing flap will be going at Vref, which is usually well below 160kt, typically between 110 and 150 kt for different airliners at different weights.

As you cross the end of the runway and pass through 50 ft, you should be at Vref.  As you get close to the surface and start to flare to land, you reduce thrust to idle at some point (the technique varies with the airplane) and as you hold it off the speed will bleed off as you trade inertial energy for reduced sink rate in getting a gentle contact with the runway.  In most jets you want to be at idle before touchdown, with exceptions.

So in the interval from when you started to flare and reduce thrust, and the actual touchdown occurs, you are coasting along, settling gently as you pitch up to trade inertial energy for reduced sink, with the speed bleeding off as you do so. At touchdown you will be below Vref, perhaps 10-20 kt.  So if a 737s Vref for a given landing weight was some number like (wild guess) 135 kt, by the time it makes ground contact, depending on how much effort the pilot put into holding it off trying to get a gentle touchdown, it'll be doing perhaps 125-130 kt.

In any case, touchdown speed is always below the approach reference speed, unless the pilot just flies into the runway like a carrier landing, which will get lots of people excited.

160 kt on short final is pretty fast for most jets, but you could be going that fast if the flaps had failed while retracted or at an intermediate position and you had to use a higher approach speed (on the CRJ200, the reference speed with flaps failed fully retracted, which they do from time to time, is about 165 kt +/-).  Same thing though; during the landing itself, the touchdown speed should be less than the final approach speed due to the energy conversion process of the flare and touchdown.

If your reference speed was 140 kt and you crossed the threshold at 160, yes you would float quite a bit if you used normal landing technique. You would also get sent for remedial training.  If you tried to drive it on at that excessive speed, there is a good chance you would have nose wheel contact first (called a wheelbarrow), which would get people even more excited.