Your questions are similar to those asked by a number of players when they first begin playing. I hope you find the answers below helpful.
Q. Can I demolish and rebuild buildings/roads etc?
A. Yes. When you click on the build button a menu will drop down from behind your Forge Point (FP) meter. Sell, Move, and Cancel from left to right. Sell destroys or demolishes the selected building, decoration or road segment.
Q. How do I research/join a Guild?
A. To research a guild: Click number next to the laurel wreath in the upper left corner of your screen. This will bring up the Ranking sheets, click the Guild tab. If you are researching a specific guild, say the “Imperial Knights”, type the name in the search box. To go to that guilds profile, click on the castle button to the right of their name.
To join a guild contact somebody, within that guild, and ask for an invitation; then accept that invitation with it arrives. Usually, the guild profile with provide you with instruction or a point of contact if you are interested in joining them. (At least the better organized ones do.)
Q. If I do not check game every day do I lose what I've produced?
A. Yes, well at least the supplies they have a set shelf life.
Q. How can I give/receive gifts?
A. Not as such. In the inventory there is a gift tab and there are gift buttons on some items that will appear in your inventory from time to time, but those are not enabled. As to whether they ever were or will ever be I do not know.
Q. How can I earn diamonds faster without having to pay?
A. The three ways of earning diamonds are as follows:
1. Purchasing them with real money via credit card, ruled out by your question.
2. Being awarded them as rewards for some quests. Those are neither fast nor predictable enough to provide details.
3. Earning diamonds outside of the game. There are number of vendors and companies that offer FoE players diamonds for various things. Taking surveys, applying for Credit Cards, etc.
Both options 1 and 3 can be access via the golden yellow plus sign in the upper right corner of your screen, next to you diamond balance indicator.
Welcome to FoE, enjoy and have fun playing the game.