These aren't the same topics as the "letter" post, and are different from each other, so really you should have started two new threads - it makes it easier for other people to find information that they might be looking for. But anyway...
1. With concrete, uncountable "the" is used when the information is shared between speaker and listener, "a" if it is not. So :
A : We've got two exams next week. A history paper and a maths paper.
B : Oh, poor you! When?
A : The history paper is on Monday and the maths paper is on Tuesday.
In A1 the information is presented as new to the listener - s/he doesn't know about the exams. So "a history/maths paper". But by A2 the information is now shared. So "the".
You'll find more on this here :
http://www.eslhq.com/forums/esl-forums/english-questions/where-rest-room-21142/
2. Sorry, but you can't say
we do sth, doing sth AND doing something. The verbs aren't grammatical you can't continue with a present participle after a simple present. Can you give a realistic example of what you mean? But if you are just asking about "and"...
And is a co-ordinating conjunction which can join any two or more items of the same grammatical class - eg :
- two nouns
fish and chips
- two adjectives
black and blue
- two clauses
I'm leaving on Saturday and I'm staying for three days.
In the case of a clause, items which occurs in both clauses (eg here
I'm) can be omitted in the second :
I'm leaving on Saturday and staying for three days.
When there is more than one item in the list,
and is only used, but must be used, between the last two items in the list. The others are separated by commas (or in the case of longish clauses by semi-colons). Examples :
We had history, maths and English this morning.
I'm going on Saturday, staying for three days, and coming back on Monday.
If your question is whether
and can be omitted in a sentence like the last example, then the answer is no. It (or another co-ordinator) must always be used to join the final two examples in a list.