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I went above 5km mark multiple times in Nepal and Peru.
I've assumed they won't let you do EBC in winter, but apparently you can. No idea if all the tea houses work during this season. It's not the best trek in regards to the views, and you can't actually see Everest from the camp. It's visible from a nearby location though. Actual camp is covered in trash and shit from all the people who go there. Another thing I heard is that if you hire a local guide they sometimes don't lead you to the actual base camp, but stop at a place that is somewhat earlier in the route because it looks the same, so make sure you actually check where you are if you do hire someone. You'll also need to get/pay for a permit in Katmandu, and there will be checkpoints along the route.
The main issue with this and similar treks is the lack of oxygen at high altitudes, so training your heart to pump blood more efficiently, as well as strengthening legs is essential. Load a hiking backpack and go up and down long flights of stairs, preferably like 10+ floors in one direction. Increase the load gradually.
In my experience first effects of altitude sickness start at about 3k meters, with full blown effects starting at about 4k. At the beginning people start feeling headaches, getting trouble sleeping. Then most get indigestion from combination of unfamiliar food and decreased oxygen supply since body redirects more blood to the brain. After 4k meters weakness sets in - it's much harder to proceed, heart is pounding and you might struggle for breath if the walking tempo is too fast. I felt the need to stop and take a few breaths every few steps. At 5k initially I had moments that felt like I was in a dream due to lack of oxygen. Combined with diarrhea, lack of sleep and physical exhaustion, it isn't the most fun experience.
Proceeding without rush to let the body acclimatize is very important. Otherwise you might get acute altitude sickness and succumb to dizziness, vomiting, weakness, etc.