Oil: I don't need the picture. What you said is fine. Consider the LU ok for the time being.
It's always a good idea to change the gear lube every year if you run a lot or every other year if you don't. The Oil is Quicksilver/Mercury Premium Lower Unit Gear oil. Being an I/O you may want to look for the notation, Mercruiser Suitable. The lower unit of a Mercruiser and a comparable hp outboard are just about identical so I don't see why they offer/may offer a different oil for you. May not. read the back of the bottle. Walmart sells the oil in the quart container in a package which contains the oil, engine adapter, hose and pump...real convenient. I boats sells Seloc service manuals. Go to the top of this page and look in Boating Parts and Accessories. You can purchase one or just rent time on the internet. I rented mine and I had it as soon as I paid for it and it worked great for me. Pages copy very well.
Woundn't shift on the water and does on land:
Mechanism, mechanism, mechanism......that's how I trouble shoot. What is different about the two? Off the cuff, nothing. Gotta dig a little deeper.
Since you are new to your boat something you need to know. Get a helper to do the shifting, engine off, and you get your hand on the propeller.
Shift into F gear. Grasp the prop and turn it CW. It should make a clicking sound. Attempt to turn it CCW. It should resist. This is the overrun clutch in the lower unit and allows the engine to drop in speed, like when slowing down from WOT, the water is turning the prop faster than the engine is running. The propeller and prop shaft will move forward and backward slightly as this happens.
N gear if free spin both ways.
R gear is usually a different gearing setup. Square cogs and slots, no overrun clutch. The cog has to be aligned with the slot for the shifter to slide into the gear. The service manual cautions that you should NOT shift into R gear unless the prop is being turned manually while shifting, or the engine is running.......they caution that the shifting mechanism (including the cable to the remote control) can be damaged.
Now shift into R gear: You grasp and turn the prop either direction while the helper shifts into R slowly. Once in gear, notice it is locked in both directions.
Did someone try to shift into R during your water testing with the engine off where you said it wouldn't?
-----------------------
You and your helper change places. Tell them what you did so that they can repeat the process. You grasp the control and without a lot of force, nudge the control toward the rear feeling the resistance till the gears get lined up and the control moves all the way in gear.
Last, have your helper go back to where the shifting cable is attached to the outdrive. CAREFULLY test it both with the gears lined up and deliberately misalign them and check for any bending or distortion in the cable end or the shifter lever and note the difference between the two as to how the control feels.
-----------------
On PTT you have no power, dead motor, no/low oil, stuck valve, frozen pistons. Follow the hydraulic lines from the cylinders back to the pump/sump. Push the up or down button and listen for the pump to run. If it doesn't chase electrical. If it does, chase hydraulic.
-----------------
Once you have answers to this tell me what happened.