By the way, my MFJ-1026 manual states: "... we HIGHLY recommend using the rear panel T/R CONTROL line input whenever possible."  Their suggestion is likely due to RF damage to their unit when switching using RF detection.

You should be tuning up at considerably reduced power, so the SWR when you begin the tune up process isn't important.  There are ways of tuning up without using the transmitter at all, such as using a noise bridge.  You can read the details elsewhere, but the net of it is that the noise bridge sits in your coax line between the transceiver and the antenna tuner and presents you with wideband white noise that you hear with the receiver tuned to the frequency you are tuning up on.  When the tuner controls are correct, the white noise reaches a very deep null which you hear in the receiver.  Palomar Engineering makes such a noise bridge, marketed as the "Tuner Tuner".

You'll need filters tuned to the two audio frequencies which are separated by 170 Hz.  The easiest way to accomplish that is to find an old hardware RTTY demodulator, most of which have mark and space outputs just for that purpose.  They're usually line-level outputs that follow the filters in the demodulator.  The old AEA CP-1 demodulator would be a good and inexpensive approach - I sold a mint one on eBay a few months back for $10.  

It's a top-of-the-line RTTY terminal unit, used to send and receive RTTY signals.  Prior to digital soundcards and free software like MMTTY, the ST-8000 was probably the best there was.  It was quite expensive, around $3,995.  You feed receiver audio in to receive.  Not sure what it required on the transmit side, but probably custom HAL software that allows a PC to function as an RTTY machine with the ST-8000 delivering FSK keying output to the transmitter.  May also have AFSK output - don't know.  Lots of info available via Google search  on the subject.  Back in the 1970's, HAL was #1 in amateur RTTY gear, and you could buy demodulator kits from them at a pretty good price.  I used a HAL ST-5000 for several years to interface my Model 15 teletype machine to my radios.

The ARRL will make a copy of an article for a couple of dollars.  I don't know if they'll do that for non-members, but check out their website and see if they will.  

Whatever you end up doing with the station ground, make sure that you tie all your ground rods together and to the AC panel ground of your house. This can protect you from unnecessary damage in case of a nearby lightning strike.

I suggest you obtain the operator's manual for the tube rig you're talking about and follow the documented procedure.  Most, if not all, of the older tube rigs don't have any external ALC indication and you are directed to tune for max output or to dip the final while advancing the load control until the maximum plate current is reached.  If tuning for maximum output, you'll be told not to exceed a certain plate current.  If the finals are properly neutralized, the point of plate current dip and maximum output ought to be the same position on the plate tuning control.  The important thing is to not exceed the suggested maximum plate current of the tubes.  

Roughly speaking, listen for Oceania during your evening hours - maybe 6 through 10 PM, with 15 or 17 meters being a good choice.  For Asia, very early morning on 20 or 40 meters.  For Japan, early evening on 15 meters.  For VK/ZL, early morning on 40, 20, evening on 15.  There are other possibilities of course, but I find these times good for me from Raleigh, NC.  In addition to propagation conditions, don't forget to consider the local time at the DX spot.  You're more likely to find someone at the rig outside of working hours at their location.  

For a reasonably priced paddle, I'd go with a Bencher.  I've tried many paddles over the years, and for some reason the Bencher just feels better and I send more error-free code with it.  You can find the black-based model on eBay for less than $60 usually.  The MFJ version is a poor imitation.

As far as books, I'd start with the ARRL Antenna Book.  Check their website - they also offer a number of other books covering wire antennas and you'll find tips on support techniques as well.  ON4UN's book on Low Band DXing is another good source of wire antenna ideas.  Also check Cebik's website for wire antenna information (W4RNL).

It would help to know more about the sub.  I have a Velodyne sub which uses a speaker-mounted piezo-electric unit as an accelerometer to feed back cone movement information to the dedicated amplifier and that one was very difficult to tame. I ended up winding several turns of the speaker wire through a 2" toroid near the speaker terminals and used several split cored on the coaxial feedback cable, also at the speaker end.  In your case, first be sure that the RF is getting in through the speaker cable and not the power line.  If possible, disconnect the speaker and listen to the audio via a pair of earphones.  If it's clean in the earphones, then it's probably coming in through the speaker cables.  The ARRL handbook recommends, in this case, an LC filter consisting of a 100 microhenry choke in series with the speaker lead, bypassed to ground on the speaker side with a .01 mF capacitor.  Of course, the choke should be able to handle the subwoofer speaker current.  You'll need one of these networks on each side of the speaker wire to the sub, assuming one is not grounded to the amp chassis.  

I've read the QST reviews of peak-reading wattmeters, and tested several here against my peak-reading Bird 43. Most of them have quite a bit of error across their frequency range.  For example, if you're putting out 1000 watts and the meter reads that on 40 meters, it may read 750 watts or 1300 watts on 10 meters.  I finally settled on the old Heath HM-2140 SWR/Wattmeter.  It tracks dead-on with the Bird 43 from 160 through 10 meters. I've also noted that the Ameritron amp built-in meters can also be considerably off across the frequency range.

You didn't mention just how high the SWR is.  The impedance of a resonant dipole will likely be higher than 50 ohms - close to 70, so there will be some SWR even at resonance when used with a 50 ohm line.  Best bet is to try and find someone with an antenna analyzer and ask them to check your dipole for you.  You could also check the SWR every 50 KHz across the 40 meter band from top to bottom and see if it's resonant somewhere other than where you expected.  Even if the resonant point is outside the band, you should see a trend in the SWR as you move across the band and you can then shorten or lengthen it a bit and see how the SWR has changed across the band.  One technique I've found helpful with resonant dipole is to cut each end maybe 3 feet longer than you calculated, and fastening the end of the wire to the rest of the wire using a split copper wire nut.  That allows you to easily make the antenna longer or shorter just by loosening the nut and moving the ends in or out on the antenna.  At first, try changing the length in 1 foot increments and observe how the SWR has changed.  

You didn't give enough info on the pot requirements for a complete answer, but here are some options I've used. First, it's an easy matter to disable the mechanical stops at the end of rotation in a pot that normally turns 270 degrees. Remove the metal back and you'll see how to do that. Of course, you'll then have a pot that has a dead spot in the rotation. Another option is to use a multi-turn pot. All Electronics and others sell 10 turn pots that have 1/4 inch shafts. I use those in a remote antenna tuner to sense the rotational position of the inductor and capacitor elements and read them out on meters in the shack. The best way I've found to do the gearing is with the cogged belts and pulleys sold by Small Parts Inc. You can accomplish any up or down gearing you need that way, and the pulleys and belts are cheap. Another thing to consider is how to limit the rotation so that the multi-turn pots (or connected components) aren't damaged if you reach the end of their rotation. I use Cherry microswitches to mechanically sense the limits, and kill the motor voltage when the limit is reached. There are undoubtedly more elegant ways to achieve these ends, but what I've mentioned is low cost and requires no supporting digital circuitry.

The first thing you should do is check on the current requirements of the FT-847 in transmit mode.  According to the marketing info, the FT-847 consumes 22 amps when producing 100 watts of output.  The 20 amp Astron supply is rated at 20 amps intermittent and 16 amps continuous.  So, you are clearly exceeding the 20 amp rating of the supply.  The next question is what happens when you do that.  If the supply doesn't lose regulation, you will be OK. If it does, the voltage variations will likely result in degradation of your signal.  If you're sure that isn't happening (get SEVERAL signal reports to be sure)you may be OK.  Just limit your full power key down times while testing.  I had a Kenwood TS-850 that I used with the 20 amp Astron supply for quite some time (the 850 drew 20.5 amps at full output)with no ill effects.  The safe thing would be to get a higher capacity supply, and you can consider one of the smaller switching supplies to keep the costs down.