I can't tell the difference between Earth Balance and butter, so that's my go-to substitute for that. I try not to eat it a ton because it's still not exactly a "health food" but it is trans-fat free unlike most butter substitutes. It does contain palm oil, but if you're concerned about sourcing you can check out the page they have on that and see if you're comfortable or not with their sourcing choices.
I really only use it if whatever I'm making needs butter though. Most of the time other things work just fine (like olive oil, hummus, or other spreads for bread). You can also spread avocado on bread!
I'm not a big fan of store-bought imitation cheeses. The ones I've tried (mostly Daiya) taste gross to me, and they're not super healthy, so I just stopped trying to replace slice and block cheese and found other things I enjoyed instead. Cheese sauces or softer spreads, however, are pretty easy to make at home using nuts as a base. There are a lot of awesome recipes online for cashew "cheese" spreads, nacho "cheese" and macaroni and "cheese." This is my favorite nut-cheese spread I've tried, which I use to make grilled "cheese" sandwiches a lot: http://www.choosingraw.com/raw-vegan-walnut-cheddar-cheese/ (I make it with cashews instead of walnuts). Don't expect it to be exactly like cheese, but it is delicious in its own way and has the comforting feel of cheese. It kind of tastes like goldfish crackers to me, which I think is awesome.
Some things that typically use cheese can just have the cheese left off altogether. I find I actually like pizza way more now that I just eat it cheeseless. Instead of it being an indulgence that leaves me feeling gross, I actually feel great after I eat a pizza with some sauce piled high with veggies. Avocados can be added to stuff for fat and creaminess, artichoke hearts have a kind of sour tangy flavor that makes up for a lack of cheese (I use them on pizzas all the time for this reason), and as mentioned above, nutritional yeast can add a sort of cheesey flavor to things.
I'm a dairy-free vegetarian rather than vegan, so I haven't actively tried to replace eggs completely in my diet (I don't use them in cooking at home though, and eat them maybe a few times a year), but like you I eat them so rarely so I've never needed to create a replacement for things like scrambled eggs because I just never make scrambled eggs. Mayo is super easy, as you can get vegan mayo at the store made with oil as a base rather than eggs. If you want a healthier alternative, I've found blending tahini (sesame paste) with water, lemon, and garlic creates a great mayo substitute. I use this recipe as a starting point for that: http://www.tastebook.com/recipes/1651089-Tahini-Mayonnaise-Eggless-and-Oil-less-
In baking, I usually go with one of these subs: http://chefinyou.com/egg-substitutes-cooking/ Different ones seem to work better for different baked goods.