First off, the vast majority of my experience is with PixInsight, however you can blend data in PS as well, it's just done differently. I'll use PI for examples.
1. What is the best method for adding Ha to an already processed color image? For example I have some Ha data of M31 and an already processed M31 from my OSC and want to add it. Do I have to go back and start all processing from scratch or is there some way to add it in reasonably well without starting over?
It is possible to blend Ha directly into existing color channels. You can just to a ratio blend, maintaining a total of 1.0/100% for the whole channel.
I usually use PixelMath for blending. In PS, you could combine channels separately in different images then paste those channels into the appropriate RGB channels in a new image. You could use layer blending to adjust the percentage of the blends.
In PixelMath, just multiply the color channel of the target image ($T in PM, and it's indexed, so 0 = red, 1 = green, 2 = blue) by the Ha channel (or, for that matter, any nb channel, or even multiple nb channels at once). So if you wanted to blend Ha into R by 70%, blue by 10%, and green by 2.5%, and OIII into B by 20% and G by 10%:
R: $T*.3 + Ha*.7
G: $T*.875 + Ha*.025 + OIII*.1
B: $T*.7 + Ha*.1 * OIII*.2
This will produce an image with the desired blend, by combining an existing color image (RGB or LRGB) with both a separate Ha and OIII channel.
2. When adding Ha to LRGB (let's say from scratch this time) should you process the Ha individually first (i.e. noise reduction, curves, etc.) or just combine it with L and R channel at the start and then go about everything else normally? What is the best ratio of combination to use? I have been using PixelMath to say something like "L + 0.8*Ha" or "R + 0.5*Ha) - not sure if that is the best way to do it.
I would generally process all channels individually first, then combine as appropriate. You can combine in many ways. You may "pre-combine" the Ha with R (and maybe B, and even G!) before combining into RGB, then combine the L. You could do the LRGB combination first, then blend in Ha into the RGB channels after.
Blend as described above.
3. For emission nebula like the California Nebula, is there any benefit to doing HaLRGB, or is just HaRGB fine? I've been looking up images of it and I am finding a lot that are just HaRGB. I ask because I have a bunch of Ha on it and want to start adding color and not sure if I need to get Luminance. What benefit would it provide, if any? Are there any guidelines for when to use HaLRGB vs just HaRGB?
I would be wary of making Ha your L, or of blending Ha into L. The Ha signal is a very, very narrow emission, whereas L is the entire spectrum. The two will often, and usually do, represent very different structure. Blending a little Ha into L could work, but replacing L with Ha entirely will often result in some strange colors and strange structure, as the Ha dominates the detail and structure perceptions, and will often NOT match the full structure across all color channels. A great example of this that is easy to demonstrate is to try to use Ha as luminance on M42 (Orion Nebula). There is a good deal of blue and even green (which ultimately makes grayish color when combined into RGB) in that object (especially Running Man next to M42). When you use Ha as L, the whole object takes on a very strange appearance, and the detail clearly doesn't match the color.
Ha is just one emission of excited hydrogen gas. The gas actually emits in several bands, the two most prominent being Ha and Hb. I think a better way to take advantage of Ha is to blend it into R and B, more into R and less into B. And you may even find that blending a small amount into G helps. The key is that you control the ratios at which it is blended in, so it's not full force in every channel. This can improve detail, as well as color.