I spent last season at Espace Killy, though mostly on the Tignes side. EK is snow sure because it's so high up, but Jan 27th to Feb 4th is an excellent time to be there because you'll definitely have a great base built up and it's not a week the French has school holidays (which start the week after). Pretty much the only time we encountered lines was during those holiday weeks, and even then, the lift system is pretty well designed and distributed so traffic dissipates quickly and efficiently. There will probably be lines at the base right around 9 in the morning when lessons start, but that'll dissipate in 30 minutes.
The place I most often encountered lines was the Solaise Express chair, and that's being replaced with something higher capacity next season so it shouldn't be an issue as much. The terrain variety is huge, both on- and off-piste. Even if you were to stay on-piste, it would be a huge challenge to ride every run even just once. Definitely visit both glaciers and check out Tignes as well.
The bus network in Val is great (every 5 minutes). It's ok in Tignes at the main resort (Lac <-> Val Claret, 15 minutes). There's a bus to take you back to your own resort just after lifts close, but otherwise you're better off skiing between the two resorts. All shuttles are free.
TDC is excellent, and I had something like 20 days of instruction from them with 5 instructors last season. On-piste, off-piste, they were great and highly recommended. They're more expensive than ESF or Evo2, but I feel you get what you pay for.
Off-piste works differently than in North America, in that only the runs are avalanche controlled. As soon as you step beyond the piste poles, you're responsible for your own safety. I think we had two avalanche deaths last year. A few years prior there was a fatal avalanche some 20ft away from a groomed piste. Until you have your own avalanche gear and know how to use it, I would keep to the piste unless you're with an instructor. TDC provides all of the equipment and are serious about your safety off-piste. You can't say the same for the ESF where we saw many cases of a big group without any gear whatsoever led by someone from the ESF who would shoot off first and leave their students to fend for themselves in avalanche country.
Off-piste lessons are very dependant on the snow. Someone mentioned that there aren't as many powder days as in North America and that's true. If you time it right, it'll be magical. If not, it'll still be good but very different than in fresh. The instructors know where to find good snow even when it looks like it's been skiied out (which the easier stuff will be by like 10am). Definitely morning lessons. Better snow and you won't be as tired. The more tired you are the more your technique suffers. Remember also that EK is a high alpine resort, with terrain above 3000m. You may need to take it a bit easier the first few days.
As for lift etiquette, there's usually not anyone directly traffic, as it were. Those of the lifts are quads or bigger (6s are very common). There aren't any singles lines, so you may as well move up together, especially if it's just the two of you. I've jumped on a chair at the last minute plenty of times and noone has said anything. Expect to hear more English than French, for better or worse.
For starting and ending your day, depends on if you're doing a lesson or not. If you are, you won't have much choice. On your free days, I would head out as far from the base of Val to explore. Head out to the Val glacier first thing one morning, for the best views in all of EK. Great intermediate skiing there. Then start to head slowing back and spend some time along the lifts you encounter. Spend another day going up to the Tignes glacier and lifts around there. Another day exploring down to les Brevieres in Tignes. As long as you end up back on the Val side when the lifts close, you're golden. The home return to Val is something of a weak point for many people. Coming down from the Belevarde your options are either Face, which is a steep black run, or the blue Santons which is often closed in the afternoon and really busy when it's not. Returning to la Daille and taking the shuttle is often a better choice.
Val is the main base for TDC, so they should be able to match you up with a group near your abilities. Call them, they're friendly. Enjoy your time off-piste with the instructor, and there's tons to ride on-piste as well.
Edit: We had our share of low-visibility days, but usually you can head up, or down, or to another part of the resort to avoid it. Check the webcams.