Should the patio heater rod wobble

Most HVAC's are designed with vents near areas or low insulation or drafts, which are typically windows and doors on exterior walls. I'd suggest the same should be done for any space heater, baseboard, or in your case, a wall heater. The reason is to equalize the temperature in the room and counter-balance any drafts.

That said, a wall heater will have different requirements from an HVAC vent, the main one being that you only have a single heater rather than lots of vents. You have to consider the power/fuel source and how you'll run the line to your selected wall. Be sure place it where combustible materials won't be placed near the heater (the specs on the heater will tell you how much clearance it needs).

One last consideration is to maximize the space that can receive heat, and in particular the spaces that you plan on using the most. So if you place it in a room and close the door, no other rooms will be heated, but that room will be heated really well. If you place it in a hallway, the whole home will get some heat, but each room will only get a little of that heat and only when their doors are left open.

I usually encourage DIY projects, but when it comes to gas applications, I have to strongly caution you to consider using a licensed gas installer to connect an on demand water heater. Nothing in a home is more dangerous than gas and it MUST be installed correctly and pressure tested. Venting the exhaust is extremely important as well, and a mistake can be fatal.

An on demand water heater is a great idea as long as it is sized correctly. I don't thin you will be happy with a 20 or 40 pound tank however. Your local propane dealers will usually supply you a larger tank at no charge as long as you buy the gas from them. The other advantage is that they will come and swap out spent tanks or refill them on site so you don't have to be lugging a tank back and forth to a refill station, and re-priming and firing the system every time you change tanks.

Bottom line is that if you are not experienced with gas installs, don't take a chance with the health and safety of your family. You are not hooking up an outdoor BBQ!!