So do you think that people can grow close to God outside of the Mormon Church, even if less easily?
Of course they can.
Is there any difference between an LDS church and an LDS Temple?
Churches, or meetinghouses, are where we hold our weekly services and other things - just churches. Anyone is welcome to be a part of anything we do there.
Temples are for particular rituals, and there are roughly 130 worldwide. You need to be a member in good standing to go to one.
Also, what happens if you lie on that question? What kind of penalties will be incurred, either in this life or the life to come?
Well, it's basically hypocrisy, and lying to God. You might get away with it in this life, but it's not exactly the sort of thing that would bring you closer to God.
Also, if you decide to drink tea one day do you automatically lose your Mormon membership, or do you have to be unrepentant as well?
Few sins lead to excommunication - but any sin, if you are unrepentant about it, can lead that way. Drinking tea would (for us) be like lying, or being rude, or something.
Why would God have no problem with his Son and closest disciples drinking something, but he has a problem with his church drinking?
God has given different rules to His people at different times. As we live in an era of municipal water treatment and understanding of the germ theory, we have access to many more beverages besides booze that we can drink without getting sick.
Also, how do Mormons do communion, if at all?
We call it "sacrament", and it is the key part of our weekly meetings. It is prepared and administered by Aaronic priesthood holders (young men, usually) and consists of two parts - bread, and water. These are put in small trays and passed out to the congregation after a prayer is said over them, first the bread, then the water.