from the photo on the web site the K3 gets its DC via Powerpole connector. If so, you must have an external power supply that has a Powerpole connector on the end of the cable. use a Rigrunner for your distribution, and no relay needed - when you turn on the external power supply, power goes to all the units on it. You should be turning off the K3 before your turn off your power supply anyway. No need to run all the accessories through the switched 12v jack on the back of the k3.
If you don't need the fancy fusing on the rigrunner, just go with a powerpole junction connector - I have a 4-way unit I use with my HF/QRP in the park unit, with my solar panel, Gel Cell, and radio.

In addition to a boom mike, you might also want to get a foot switch, so you won't have to use VOX.
There are some inexpensive single headsets with boom mikes - check the ads in the back of QST - the one for www.warrengregoire.com jumps out at me in a quick scan., or look at your local ham store if you have one.

Check out the possibilities at www.globalindustrial.com - wire rack shelving and desk options.
I am using a unit about 6' wide and 6' high, with a desktop surface the full width, and a comfy height for computer keyboard, laptop, keyers, keys, etc. One shelf above the floor, with slide out bins for coax and computer paper and such, and three shelves above the desktop, for accessories, manuals, etc. The whole family tech area is on it - my side has ham gear and books, my laptop, the other side has tower PC and big monitor, the printer/scanner, network hard drive, router with wifi, etc. It cost nearly $800 with all the accessories - side rails, and such, but it's been ideal - and sincie it's open on all sides, it's easy to get to cables and such.
I don't see my exact unit on their web site now - the closest models are mobile, have big casters. Ours has no casters. If you go for something like this, do NOT use the pullout keyboard tray - put the work surface at a comfortable height to begin with - ours is 27" from the floor.
You'll do best with their hardcopy catalog, much easier to find the basic model you need, and see the accessories.

I've found that some SO-239 double female connectors lose their grip after a number of years. Your lightning arrestors likely had similar problems. When you have one coax run feeding one antenna on several bands, and have issues on one band, the problem can also be that you have an odd multiple of 1/4 wave in the length on that band. Add a foot or two using a jumper and see if the problem goes away.
Also - with feedline junctions out in the weather, be sure to carefully waterproof them.

doing CW with a FM rigs, MCW is the only way to make it two way comm.  there have been some articles in QST on simple setups to make the keying less clumsy than sending on the DTMF pad.

If the conduit isn't straight, then large diameter cable will be difficult but you need the big
cable for the VHF/UHF side. LMR400 Ultraflex ?  how tight are the conduit corners ?
for the antenna switch - I've had good luck with the Ameritron RCS-8V, but am using all five ports -
you might go with the RCS-10V for 8 ports. I used indoor/outdoor Cat3 cable for the control run
with my 8V. But keep in mind that with a switch, you can only use one antenna at a time -
so monitoring a repeater while working HF is going to be tough. You might even thing about diplexers
at each end, if you can weather protect the one at the roof.

If the mag loop works on the balcony, if you have the budget, get a remote tunable magnetic loop - MFJ makes two of them - one for 40-15m, the other for 30-10m.

when you have odd problems on one band on a multiband antenna, the problem can be that the feedline is an odd multiple of 1/4 wave on that band. Try adding a short piece of coax (1-2') to the feedline on the shack end of the cable,and see if the problem disappears.

If the vhf vertical has built-in radials (a ground plane) or doesn't need them (like a j-pole) what matters is how far the antenna can 'see' - how far is it to the horizon.  If you raise it to clear a building or other nearby obstacle (a hill), it will make a difference, otherwise height won't matter very much. High is good, but a lot higher won't be too much better.

take up falconry as a secondary hobby ?

What's best depends on your budget. Mine was/is limited - I had a 3 el 6m beam and a 14 el cross polarized on 2m, on a cheap TV rotator.
Didn't want to spend $400 on one rated for the load. Cheap rotator didn't point reliably, failed, replaced, repeat. we have santa ana winds out here in SoCalif, and I was lucky the whole stack didn't come down in 10 years. and with the beam pointing not reliably, I missed some contacts. Took down the beams and rotator, replaced with phased Omniangle Loops for 2m,and a single Omniangle loop for 6m. works great! Now I've finally worked some grids in Washington state that I could never hear before because of nulls on the beam.  I have no nulls, and the antennas are never pointed in the wrong direction!
I'm not going for VUCC, just like to make contacts in a contest when the band is open.

Fred, KQ6Q

Don't bother with a tuner for 2m and 440. Use the best grade of coax you can afford. The bandwidth of antennas for those bands is such that no adjusting of the antenna is necessary, and if there was a mismatch, putting a tuner at the transmitter end would not help. Just connect the antenna and enjoy.

Fred KQ6Q

If the Arrow J isn't doing the job for you, then you need to look at other factors - how high is it ? can it see the horizon? In what directions ? What kind of coax are you feeding it with ? how long is the coax ? What kind of radio are you using? How far are the repeaters you're trying to hit ? Can you hear them clearly ? Have you tried simplex with a local ham ?

PGT - the fact that you can hit the 2 meter repeater of the club with the 10 meter net isn't relevant - it's on a tower or tall building or mountain peak. Unless they're using a 10 meter FM repeaters, they're working from their home stations, with horizontal or vertical antennas. If they're using vertical antennas, and you're using vertical antenna too, you'll either hear them on groundwave, or not. If they're using horizontal antennas, you might hear them, but likely not. polarization matters. When I was a new novice, my elmer buddy lived about 20 miles away, used a vertical on 40 meters. I had a dipole on 40 meters, and he was off the end. We could NOT hear each other at all. Just the way it was on 40. We could have worked on 2m if we could have seen the same repeater.

You really ought to consider getting a real mobile radio, with a remotable control head. Using an HT while mobile, even if mounted and with a headset with mike, is going to be distracting. You could put the radio CPU in the trunk near the antenna, and just use the link cable to reach the control head in the car. the display will be much larger and brighter, the controls easier to use, and you'll have a LOT more power, and not be worrying about the rechargeable battery.