Over the years I have converted more than a few 9.9's to 15 horse (and 8's to 9.9's etc).
Normally I would tell you not to bother because the cost does not justify the minimal gain, but since you have all the parts, you only need to pick up a couple of gaskets and start swapping parts.
First off - the carb(s) will bolt right up back and forth on all models built between 1986 (first model year) and 2012 (last model year - sold outside of the US only after about 2005). The "part numbers" changed over the years, but they are all back/forth swap-able.
Second - the exhaust tube is "integral" to the exhaust plate - not saying you couldn't jury rig it, but the tube was never sold "by itself".
Now there was a change to the water tube and the exhaust plate in late 1994 BUT they will still "bolt right up" to earlier and later models.
With the early plate (and tube combo) the water tube had a flare at the top and with the use of a (clip) it was screwed to the plate which kept it in place - the tube itself was a constant diameter from top to bottom.
After the change the tube simply attached to the plate with a grommet. To keep it from falling out when the lower unit was removed they placed a flare in the tube about halfway down. Since the flare prevented the tube from passing through a hole in a cross-member in the exhaust housing (leg - the flare was "above" the cross member) the tube is essential trapped in the upper half of the leg.
So - if you are using a pre-94 1/2 exhaust plate AND water tube it will fit without issue on a (post) model. The water tube fits through the hole, the plate fits the motor and the tube extends with no issues into the exhaust housing.
IF you try and put the "post" plate and water tube on a "pre" exhaust housing you will find that the water tube will not go down far enough. The cross member (support) in the leg is "higher" than on the later models and the "flare" in the tube, meant to trap it stops the tube from passing through enough for everything to line up properly. You can try installing the water tube from the bottom after the exhaust housing is attached (but that is more than a tedious pain in the butt - trying to hit that little grommet, blind) or what I did on all I worked on was simply used a long drill bit (installers bit) to drill out the hole in the support big enough to let the whole flare pass through.
The only issue with my method is, the tube is now "not trapped" and can be easily pulled from the grommet whenever you pull the lower unit.
Since your scrounging the 15 horse "parts" from the older model you should not have any issue.
Pull the powerhead on your 9.9 - remove the exhaust plate/exhaust tube as one unit (since it is) and the water tube. Use a new gasket and put your exhaust plate/tube and water tube from the 15 horse on your 9.9 exhaust housing.
Replace the powerhead with a new base gasket and then swap the carb.
Reconnect everything and fire up your (new) 15 horse.
Don't worry about the prop. Both the 9.9 and 15 came from factory with a 9 pitch. You have done nothing to the displacement so your torque band is the same, it's only in the last few rpms that you get the extra horses.
Just don't hang your 9.9 on a boat and take it out with a gps "before" you turn it into a 15. If you do you will run out and replace your gps because it must be "malfunctioning" since it may only be showing you 1 more mph than it did when the motor was still a 9.9 - and if your (new) 15 is pushing into a head wind it probably won't even make the top speed you got out of your 9.9.
But again, you can do the "upgrade" for about 25 bucks (exhaust, base and carb gasket), so why not...
