When you are looking for a cave, look in the mountains, but while in the mountains look for a bear. Once you find it look around the area it's in and you should have a cave.
Animals and Pressure Plates
Animals CAN step on pressure plates and trigger them. I don't know the specifics, but I figured this out because an angry bull followed me and triggered the pressure plate opening my door. Another thing too: Almost all animals are not capable of going through a 1x2 door.
Free elements
You can get free diamonds, iron, rare blocks, etc. by going into creative mode and placing all the blocks you need, then go in to survival and dig up the blocks you placed. You will get the blocks for free.
Houses
Dear, Yolo if you want the house to be strong you should dig deep underground and find rock or other hard substances that there are a lot of. Make sure you are able to get back above ground. Once you have enough go back up and start building. The best size is probably a 10 by 10 or bigger if you want. If you want to be really fancy you can also slap some wooden flooring to the house.
How to make a mansion very easy in Survival
What you do is modify a world to creative then make your mansion then go back to modify and turn it to survival (any mode) then there you go you have a mansion.
A cave the easy way...