1.

Roughly, think of "a" as meaning "whereas" and "но" meaning "however".

Here is a more detailed explanation: http://masterrussian.net/f15/%D0%B8-...32/#post127497

2.

In Russian, each clause of a composite sentence should be separated with commas.

Мы взяли зонтики, потому что боялись, что пойдёт дождь.
Эта книга не такая интересная, как вы думаете.
Ящик загораживал дорогу, так что не нельзя было пройти.

3.

Short adjectives normally used as predicates:

Эта комната занята. - This room is occupied.
Я согласен с вами. - I agree with you.
Она очень счастлива. - She is very happy.
("to be" in present tense omitted as usual)

The problem is that the rule is very irregular. There are many cases when a long adjective in preferable. Cf.:

short: Эта комната занята. - ok
long: Эта комната занятая. - wrong
short: Этот фильм интересен. - too bookish
long: Этот фильм интересный. - ok

I guess, short adjectives tend to be used when we speak about temporary conditions ("the room is occupied" now, but not forever), and long adjectives tend to be used when we speak about inherent features of something ("the movie is interesting" - the feature of the movie). But, again, there are many exeptions from this trend.

4.

When a demonstrative pronoun is used as a modifier, it agrees in case with a modified noun:

Эта книга интересная. (nominative)
Я прочитал эту книгу. (accusative)
Мы идём уже три часа по этой дороге. (prepositional)
Идея этого человека очень интересная. (genitive)

When a demonstrative pronoun is used instead of a noun, it changes in cases as an usual noun:

Это было странно. (nominative)
Я знаю это. (accusative)
Он к этому очень стремился. (dative)

Think of pronouns as of a kind of nouns or adjectives. When used in place of nouns, they change in cases as regular nouns. When used as modifiers, they undergo case agreement.


5.

There are no strict rules when you should drop a pronoun used as a subject, however there are some trends:

Normally, a russian sentence has structure topic + comment:
Я уезжаю в Москву завтра. - "я" is a topic (i.e. what is your sentence about) and "уезжаю в Москву завтра" is a comment (i.e. information your sentence provides).

You cannot omit a pronoun if it is in the comment:
Завтра в Москву уезжаю я. (а не кто-то другой) - in this sentence, "я" is the main information, so it cannot be dropped.

You may omit a pronoun if it is in the topic and the topic is clear from context:
— Что ты делаешь завтра?
— Поеду в Москву по делам.

When you have no context (i.e. just starting conversation) or change topic, you should keep a pronoun:
Я пойду купаться в озере, ты идешь со мной? - "Я пойду" - the beginning of the conversation; "ты идешь" - the topic changed.

Well, above is a very rough rule. Different types of negative and interrogative sentences have their own rules on this point. At the same time, the spoken language has less strict rules, as we can use intonation to place emphasis.
So, in general... when in doubt, don't drop a pronoun.