White wires are your neutral. You don't switch the neutral, you switch the hot, otherwise, contact with an off fixture could electrocute you if you contact a ground. Therefore, the whites are frequently tied together, with the exception of cases where the white is being used for something else, like a switched wire.
For a 3-way switch, you need an extra wire, which I prefer to use the red wire of a 3 conductor cable. It looks like they've done exactly that in your setup. For more details on how 3 way switches are wired, you can review the various diagrams on this site which include many of the possible wiring variations.
Edit2:
Thanks for the picture. That, a nights sleep, and your description of the other side of the 3 way switch narrow down how this could be wired. I was hoping for the best from your previous electrician, but it looks like you got the worst. Someone really didn't know what they were doing in this J-box. Also, that doesn't appear to be a 3 way switch on the right, at least, not yet, which is why it never worked right. Note, unless it's specially designed to work with 2, you should only have 1 dimmer on a 3 way switch configuration. The other side should be a non-dimming 3 way switch.
Here's my assumption: Bundle 1 goes to fixture 1, Bundle 2 goes to fixture 2, and bundle 3 goes to the other 3 way switch. It looks like all the line wires are in the fixtures themselves (otherwise, you'd have at least one more bundle coming into this J-box).
What I'm not going to assume is that white is actually white in bundles 1 and 2. Most likely it is because of how the current switches are wired, you want to test this to be sure. With the power off and the old switches disconnected, check for continuity between each of the wires in bundles 1 and 2 to ground (use bundle 1's ground for your tests). And with the power temporarily turned on (be careful with exposed live wires), make sure the current only appears between ground and black. If any of the wires are mis-colored, put some tape on the end and label it, and go by the label, not the color, in the below instructions.
Here's what you should do to clean up the mess:
Step 1: Tie all the grounds together, including all 3 bundles and both switches that you replace. Use a short piece of exposed copper to connect the switches if they don't have their own wires. Make sure the wire nut is large enough for all 5 wires. If you can't find a ground on bundle 2, I would double check fixture 2 to make sure it's grounded, and if so, the ground from bundle 1 is all you have to rely on for everything else.
Step 2: Wire up switch 1 like normal, black and white to the two screws on the switch. Nothing connecting the other side is necessary if the above tests worked. If the white from bundle 1 is faulty, connecting to the white in bundle 2 will work, but you really want to find the problem and fix it.
Step 3: Your 3 way switch appears to be variation 1 on the above site, with the line coming to the fixture itself, bundle 2 is the left line in the diagram, and bundle 3 is the right wire in the diagram. The whites from bundle 2 and 3 are tied together and to nothing else (unless a connection to switch 1 is needed). The black from bundle 2 connects to the typically white screw of switch 2. The red and black from bundle 3 connect to the two separate black screws on switch 2, but it doesn't mater which is which at this point. Double check the other end of the 3 way switch to be sure that the white is on the correct screw.
Step 4: Temporarily check your work with the power on. If you don't like the direction of the 3 way switches (I prefer both switches being down as "off"), you can just swap the red and black on either switch to reverse them.
Tip: if the dimmers are large, and the push in connections on the back have an adjustable clamp that closes when you turn the screw, then use the push in connections on the back of the 3 way switch to reduce the risk of a short on the side.
Standard warning: while working with electrical, always have the power off unless you are doing a quick test. Before doing any tests, make sure no wires are shorted with any others. And during any quick tests, make sure you don't touch those wires with anything other than the voltage tester.
OK, here goes:
What you have now
The power from the panel seems to come in in the middle of the bottom of the box. That black wire is always hot, and is connected to all of your switches via the orange nut you labeled e, which connects wires d, g, i. i connects to j via switch 3, and j (which is the same as m) connects to the always-hot (k) via switch #4. You shouldn't use the side screws and the backstabs at the same time (you shouldn't use the backstabs at all), especially since it looks like those screw terminals will accept two wires each.
a, f1 (f3), L, and h all look to be switched-hot, and will only be powered when the corresponding switch is in the 'on' position. L is a little troublesome since you said it only controls 1 fan, yet there seems to be 2 wires there. A clearer picture of switch 4 might help.
What you need for the new dimmers
Essentially, keep what you have. Each switch gets wired to the always-hot via a black wire. You can run a pigtail from each switch and nut them all together with the always-hot, or daisy chain the always-hot from one switch to the next. A lot of switches and dimmers come with pigtails already, so that's probably the easiest. If you do daisy chain, use the screw terminal for both wires, unlike how it's wired now.
All of your neutrals (white) get nutted together. There's a big bundle of neutrals already nutted together in the back of the box. Your new switches may require a neutral, in which case you'll need to tap into that big bundle. Given that there's already 5 wires in that nut, you'll probably have to separate it into 2 or more bundles and pigtail them to each other. Or use some large (8-port) push connectors instead of wire nuts. If your new switches don't require a neutral, leave that big bundle alone.
All of the grounds get nutted together as well. It looks like none of the switches currently has a ground attached (which is a no-no), so you'll need to add pigtails for those, and bundle them all together. Make sure at least one pigtail remains attached to the metal box, as well. Looks like you could pigtail the grounds for all 4 switches together with a pigtail to one of the screws on the box behind switch 4, as the rest of the grounds seem to be attached to the box already.
Finally, your switched-hots (a, h, f3, L) need to be connected. Get some electrical tape, and tape those wires with red or blue (or something other than white or green), so you can easily identify them in the future. Attach each to the corresponding switch. If there are multiple wires, use a pigtail and wirenut, or put both into the screw terminal (if the terminal supports it).
Best Answer
The cure for this is a surface conduit starter box. This box will sit on top of the existing box about 1-1/4" proud of the wall. It will provide both the statutory cubic inches you need for all those wires and yokes, and the practical cubic inches for your dimmers.
Normally it's intended that you then use its sides as a launch point for surface conduit, but if you don't do that, I won't tell :)
Now, I can't guess what's going on with all those black and white wires. I surely hope you can. Normally I expect to see some of them be wire-nutted to each other, and this positional data is very important. Usually the only evidence that tells you which wire does what is how they are connected in the box. I gather two of the blacks are switched hots, and the other two are supply and onward, which need to be connected together obviously.
With those ground wires, I would tuck 3 of them back, and send the fourth into a wire-nut to meet the pigtails from each switch.
Now as far as cubic inches statutorially, you need to count the wires and the "wires".
I see 8 wires in there - 4 pairs of black-white. I also see a bunch of grounds - all grounds together count as 1 wire. If any of the switches have pigtails, pigtails do not count. Each switch (or to be more precise, yoke) counts as 2 "wires". So I see a total of 8+1+2+2=13 "wires" in the box.
#14 wires require 2.0 cubic inches each, for 26 cubic inches. #12 uses 2.25 cubic inches per wire, or 29.25 cubic inches. The 24 c.i. plus a starter conduit cover should get the job done.
If you want flush mount, you must replace the junction box in the wall. What you want is a 4-11/16" square box with a 2-gang mud ring of appropriate depth. This will be bigger than your normal box, but will undoubtedly fit the two large dimmers. You can try getting it done with a 4x4" square box with a 2-gang mud ring, and this will fit in the footprint of the existing box... but it could be a tight fit.