Here are my problems:
My joystick does not come off.
Your best bet is going to be to talk to the dealer and if need be the manufacturer and/or importer of the chair. They'll be able to inform you the best about your chairs options. Having said that, I would do everything in my power to find a way to have that joystick off. That or at least something within the circuit that can not be overruled as long as that specific part isn't at hand (a key by the way does not help, they can be bypassed in about 1 second

). If some nutcase does decide to want to drive, reconnecting the battery is way too simple.
My backrest has two BIG handles on it, which are great for grabbing but my back does not lock down, so pulling on it will do MAJOR damage.
Tape them down, or fixate them to the back in some other way. Whatever way that keeps the chair in tact but does prevent them from noting them and using them for their convenience. That's a lot more preventive than any note telling them not to. Not being able to makes sure they don't.
You mean back is not locking down after being folded? Very few do. Mine doesn't. That's not a real biggie as they'll 'lock' instintively if the backrest moves while in flight. It'll have some range of movement but not beyond OK. As far as airport staff etc go; they tend to push a backrest down, not pull it up. Another helpfull thing. As long as they can load your chair in an upright position, they tend to not deal with the backrest as long as you'll provide enough clear info on where to lift the chair. If for your own piece of mind; you can use anything as simple as velcro, some rope etc. to keep the backrest from moving beyond whatever point you want it to. Tie it down to the seating and you're done.
I am taking off arm rests and head rest. I am taking off the armrest with the joystick and wrapping it up and basically buckling it into the seat, so it cannot be bumped easily.
Oy. Leaving a joystick on the chair but after removing the arm rest it's attached to from the chair. That's a risk of the part getting lost and if you can not do without one part it's the joystick. Things shift allover the place during a flight down in that container and at the airport. It takes just one employee not aware of the fact that the armrest+joystick fell of and is in that one dark corner and big problem. Don't know how bulky or not the armrest is, but if you do take it off I'ld look into taking it with you as carry on. Another thing with joysticks is that the material used to cover certain touch-buttons can be damaged easily by any 'sharp' corner, that kind off thing. I've seen it happen before. If you have it on you, you have an eye on it yourself.
Have it in the carry on, leave it on or tape it to the chair 100% so even you will have a difficult time getting it back of, so to speak.
Tilting it will NOT make my chair shorter, and I have no intention of telling them it can tilt.
You do not need to. By tilting they do not mean tilting like you would tilt the seat. In airworld tilting is putting your chair on it's side. Or diagonally. Upside down for all they care if it makes loading easier. And that of course seriously increases the risk of damage. That's why it's so important to have the chair being low enough it's very easy to load standing up, not even giving them the IDEA it might need tilting.
You're thinking too much like a powerchair user. Well, duhuh, GLLS! I know.

But they don't think like one. They know squat about the options a chair can come with. Most of them will not even know of a tilting-seat option. They think of the chair as a very heavy, big piece of luggage. When looking at it from that perspective, you can imagine they tilt by tilting the chair, not the seat.
Pretend you know nothing about mobility aids, but see that heavy thing in front of you. You need to handle it and get it onto a convined space and will be under pressure time wise. You can see how tempting it is for many to go for the easy way, even if that's not always the smart way. Thinking like that is a great tool for seeing any possible 'weak spots' of the chair for flying and thus preventing damage.
Really, these folks do not mean any harm, to do wrong or anything close to that. Those you can talk to most of the time will mean very well if not beyond that. It's just the lack of knowledge that can result in them not seeing a risk you do see.
I am going to bring both chairs through security to the gate, fold down the back there and explain how to push it, and they will be able to get it on the plane (it is less than 36" tall that way, 39" long, 26" wide).
Remember that you talk to one individual, but more than one will touch the chair during loading and unloading. Verbal is good, but always have it in writing on the chair also. Short off course, long texts will not get read in the hurry of things. This way the folks dealing with the chairs you haven't talked to have the biggest shot of being informed.
The tie down spots are very clear (I have a transit kit). I am making them with RED duct tape.
Remember to not think from our point of view.

Some of these folks can be like "can you point me out the tree in that forrest?". There is no overdoing it on this department.
I am giving them my powerchair at the top of the jetway after I strip it down and cover the handles with a pillow case over the chair back (to protect it from getting damaged too).

Perhaps you want to cover those before that stage. By then somebody could have seen them, giving them the idea of using them in case of some troubles with loading. As long as covering those handles doesn't interfere with you using the chair, I would do it before hitting the area staff can see it.
I am going to bring my manual plane-side and fold down the back there and give it to them (I will have stripped off removable parts before that). I would prefer to see both of them when I get off the plane - the manual at the plane and the powerchair at the top of the jetway. Both will have gatecheck tags, so they SHOULD do this.
Should, could, would, theory isn't always practice. Especially when flying with 2 chairs, it does happen enough times one chair will be down the runway and the other at the gate or bagageclaim.
This will hopefully work - Since I cannot take off the joystick I am a little stuck, but I am hoping that if I basically bubble-wrap it and impress upon everyone how expensive it was, they may listen.
You're trusting folks that do not understand what the chair means.

Money doesn't make them blink an eye. If somebody does understand; great! But when dealing with these situations you should act from the idea that folks do not. That way, you're protected if need be.
To illustrate how far beyond idiotic that 'do not understand' can go, I'll take you back to 2008. I flew to MCO with my 2 chairs. Marrechaussee did some damage on purpose to my powerchair, Lufthansa did a lot more -reason unknown

-. They offloaded my manual chair, because somebody was trying to use their brain. The papers said; load 2 chairs, but the brain said "one person, one chair should be enough, this one can get out of the hold". Result; one damaged powerchair -that died later in that trip- and one manual chair left at my home airport. After some dumb moves by LH, the manual chair finally gets into MCO about a day and a half later than could've been after discovering the missing, on a UA-flight. Did I get my chair? No.
Rules of LH and UA dictate that
bagage coming in on the flight of one of them, ment for a pax scheduled on a flight of the other one has to go through a process before released to the airline the pax was flying and then being released to the pax. This would take about 24 hours. Both LH and UA were deadserious in stating my manual chair was at MCO, in a UA-office about 3 offices down from the LH-office and it would stay there for 24 hours. That's our bagage-policy, mam. No less than 5 members of staff (UA and LH) did the same drill, immune to my "it's a wheelchair, not a lost suitcase with some replacable clothes, I NEED it to get around, you've got my legs sitting around in your/their office". Somehow giving them the phone number of my lawyer did the trick and 90 or 120 minutes later the chair showed up at WDW, but that was after fighting a brick wall for way too long, making way too many phonecalls etc.
They don't get it. As long as they aren't dependant on one themselves, the do not get it. Most are sympathetic, but alas that's not the same.
An a different note; also remember that you are allowed to have the tools in your carry on needed for those chairs. Have them in the carry on. If a bolt comes loose or whatever, you can fix -or have somebody do it- the problem on the spot. Nothing worse than having such a relatively small problem but being stuck because you haven't got the right tool handy. Good thing is that many brands use the same types of technique, resulting in a good shot that you can use the same tool on both chairs. I travel with about 5 or 6 different tools and with that any and all "tool-solvable" problem with any of my aids can be dealt with. A God sent! Same thing goes for the battery charger. I know it's a pain, especially if yours is an unbelievably heavy one. The rental I use for the US is.

If something does go wrong with checked baggage; you'll at least have the charger with you and wont be stuck with only x miles left to go until somebody saves you by either finding your lost luggage or a new charger.