First things first. Every camera's AF system is a bit different. The shape and size of actual areas of sensitivity for each AF point are not the same size and shape as those little squares in your viewfinder that represent them. They're usually much larger and often aren't square or symmetrical, particularly the AF points on the periphery which are very often "L" or "T" shaped.
A lot of what gets blamed as poor AF performance is attributable to operator error. If the "focus point" in the viewfinder is placed over an indistinct target with very little contrast, and another area of higher contrast is within that AF point's entire area of sensitivity, the camera will almost always latch on to the area of higher contrast. If that area of higher contrast is closer or further than our intended target, we accuse the camera of front or back focusing!
In order to test and adjust autofocus, we must be sure our testing setup is correct. The AF target needs to be flat and aligned parallel with the camera's sensor. It needs to be high enough contrast and far enough away from any tilted scales (often used to gauge how far and in which direction the AF needs to be adjusted) that the camera doesn't try to focus on something other than the properly aligned target. If the target is tilted with respect to the camera's sensor, your results will not be valid and will often make things worse.
Although not Nikon specific, here's a rundown in a lensrentals.com blog entry by Joey Miller about the basics of doing AF adjustments. Of course the menu specific instructions will vary from one camera to the next, so you'll need to consult your Nikon D810 User's Manual for exactly how to enter the adjustments.
1 - If you have entered an AF Fine Tune correction for each lens, the camera will normally recognize and remember each lens when it is reattached and use the amount of correction you have entered for that lens. The only time this may not be the case is if you have two copies of the exact same lens model or two lenses that report the same exact lens ID to the camera. Be aware that some third party lenses will spoof the lens ID of a makers lens, so if one of your third party lenses is reporting to your camera that it is the same model lens as one of your Nikon lenses (or another of your third party lens using the same lens ID), it may not be able to tell each lens from the other. Some cameras with AF Fine Tune (Nikon), AF Fine Adjustment (Pentax), or AF Micro Adjustment (Canon, Sony A-mount) can discriminate between multiple copies of the same lens and some can't. You should check your owner's manual to see if the D810 has this capability. You can check the EXIF info from images taken with each lens to see how your camera identifies that lens.
2 - Keeping contacts clean is important, but it won't have any effect on the accuracy of AF. The communications between cameras and lenses tend to be digital, rather than analog. It might have an effect on whether AF engages at all or not.
3 - Third party lenses tend to be less accurate and consistent in terms of AF performance than maker's lenses. This is particularly the case when comparing premium maker's lenses with higher end third party lenses. That's just a fact of life and should be considered when deciding on what lens to purchase. When issues do come up using older third party lenses on newer camera bodies, it usually means a feature (AF, aperture control, etc.) doesn't work at all.
There are a few other questions here that talk about different aspects of AF adjustment. Here are some which you may find helpful:
How do I diagnose the source of focus problem in a camera?
Focus point causing unsharp images?
Which offers better results: FoCal or LensAlign Pro?
Do the issues with sharpness I am seeing require AF fine-tuning?
How can lens cause consistent front or back focus?
How can I more consistently focus on the point I want?
Why does my DSLR focus on the background instead of my subject when taking shallow DoF photos?
Canon or Tamron 70-200mm F2.8
How can I fix back/front focus on my Canon 50mm F/1.8 STM lens?