[Dany]

Hello Hela,

I will start.

Originally Posted by hela

1) Would you please help me finish these clauses ?

a) If you were to give him a chance…
b) If you were to fail …

c) Would it be possible to use the indicative “If you ARE …” in the above clauses ? What difference would it make?

I wouldn't say, that your sentences are grammatical correct. Try it like this:
a.) If you gave him a chance to explain it to you, you would understand him better.
b.) If you failed the exam, you would have to resit it.

Originally Posted by hela

2) How can I make a polite request using “If + will or would”?

If you will / would ….

Please note, that it isn't correct, to use will or would in the if-part of the sentence.

If I learn, I will pass the exam. (Conditional I)
If I learnt, I would pass the exam. (Conditional II)
If I had learnt, I would have passed the exam. (Conditional III)

NOT: If I would/will learn .... or If you would/will learn ....

Originally Posted by hela

3) What do you call the mood/tense (?) that follows “would” in the following sentences ?

I will try to explain it to you, to the best of my knowledge.

Originally Posted by hela

a) If the Queen died, she would BE SUCCEEDED by her son Charles. (Bare infinitive + past participle ?)

I think, that this sentence is not correct. You have used the Conditional II.
Conditional II = If + Simple Past, would + infinitive

If the Queen died, she would succeed by her son Charles.

Originally Posted by hela

b) If I were rich I would BE LYING in the sun on a tropical island. (Bare infinitive + present participle ??)

You have used the Conditional II.
Conditional II = If + Simple Past, would + infinitive

If I were rich, I would lay in the sun on a tropical island

Originally Posted by hela

c) If it had rained yesterday, there wouldn’t HAVE BEEN many people on the sea shore. (perfect infinitive)

This sentence is correct. You have used the Conditional III.
Conditional III = If + Past Perfect, would + have + Past Participle

Originally Posted by hela

d) If I had not got married, I would still HAVE BEEN LIVING abroad. (perfect infinitive + present participle ??)

I don't think that this sentence is correct. Look at c). It's the same.
If I had not married, I would still have lived abroad.
"Had not got married" doesn't work.

The rest of your questions I couldn't answer. Perhaps another one could. But I hope that my explanations of 1, 2 and 3 could help you at first.

Kind regards,
Dany