1. Whether you use organic or non-organic methods for fertilizer or pest control is ultimately a personal choice. There will be many people with opinions on which is best, with those on the organic side tending to be the most vocal. Both methods will help you to grow a beautiful garden, but you just need to decide how those methods (or even a combination of the two) fit in with your personal views.
For example, I personally try to use organic methods, but do occasionally use non-organic methods for really tough pests and weeds. There is no real right or wrong answer to this question. It is just up to you.
As for cow manure, you can use it on lemon trees, but you must make sure that it is well rotted/composted because fresh manure will burn the roots. Also, you need to add some phosphorous to counter the high nitrogen that is found in manure.
2. If you are growing your lemon tree in a container, you will need to water daily if the temps get above 20c and twice daily when temps get above 30C. If your lemon tree is in the ground, it needs water daily for the first 3 months and then weekly after that.
I think our section on lemon trees will help you learn all about caring for this plant:
3. For the most part, a bigger pot does mean a potentially larger plant. When deciding to move to a larger pot, you need to keep an eye on how close a plant is to being rootbound. This article will explain more on that condition.
4. When you get lemons depends on a few things. The age of the tree is the most important. Trees, just like people, can't reproduce until they reach maturity. For a grafted tree (which if you bought yours at a store, it is grafted), the tree will reach maturity between 3 - 6 years. Now, that may not mean from when you bought it. The tree you bought may already be 3 or 4 years old and could start producing this year. But it may also only be a year old and you will need to wait up to 5 years to get fruit.
Good balanced nutrients, plenty of water and a healthy disease and pest free tree will help decrease time to fruiting. A tree that feels "safe" expending resources on fruit because it has plenty will produce much sooner than one that does not.
That being said, and because you are new to this, I will remind you than when flowers appear, they need to be pollinated in order to produce fruit. Just take a small paint brush and swirl it around the inside of the flowers when they appear. This will pollinate the flower. But don't over do it the first year. If the tree produces too many fruit the first year it fruits, it can harm the tree in the long run. Just pollinate a few flowers the first year and if the tree looks overly laden with fruit when they start ti grow, thin the fruit to just a few.
We do not mind the questions at all. We are always happy to see new people excited about gardening. Please feel free to stop by and ask us any gardening questions you may have. :)