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.
No way from here to know which wire fed the light or fan, but you have a continuous hot feed into the box that fed the receptacle, light and fan. The third wire, maybe it fed the fan or maybe something something else farther down the line, if the latter that needs to be tied in 100 percent to the hot feed. All whites still tied together, blacks to the dimmer from the light and hot. All grounds tied together set with a screw to the metal box. Securing the ground to the box is not needed if the box is plastic.
Before I got too wordy, here is a sketch
Still have to learn to format better
The sketch is a little rough, it should get the idea across I hope
Best Answer
All grounds should be connected together. You can make pigtails from box screw, each switch and connect it all together with feeder ground. If you using wire nuts, they have different sizes and capacity usually written on package. Just choose properly size, it is going with number.