Would

(1)

Is would the past tense of will?

Well, it depends how you look at it. You might argue that would is one  modal and will is another, and that neither of them actually has a "tense" (perhaps a "mood"?)

On the other hand, we use would as the past tense of will, so we might as well agree that that is what it is.

Would, then, may be considered a form of past tense. When I was little, I would always wake up in the morning expecting to discover that all my life had been but a dream and that I would have new parents.

The first would is definitely a past tense. The second would...harder to say, but, yes, it's just another  past tense.

We use would in this way to refer to a habitual past action, and it may often be used instead of used to. When I used to work there, I would catch the 5.15 train every day.  

(2)

Is would a conditional?

Most definitely. When not a past indicator, it is a conditional. Like could, it should therefore be treated as a conditional. What this means is that would sits happily in the clause that follows the if-clause.

Even if I had the money, I would not buy a polluting  Volkswagen,

If you were a nicer person, maybe you would have more friends.

If they had worked, they would have done better.

If you  were as Zen-inclined as you pretend, you would not have knifed the traffic warden.

But we do not always write out or say the if-clause. Very often, we leave it implicit - the if is to be understood or inferred from the context.  Yet IT IS STILL THERE. It may be hidden from sight, but it is lurking somewhere in the shadows. 

So if I write, I would like a coffee, please, the invisible if-clause would run something like: if that is not too much trouble for you. 

We would love to stay with you guys! (if you invited us, which maybe you have; if we can find the time to visit, which maybe we can't, since I am not being sincere here).

You would never think he was a poet (if you bothered thinking about it; if you considered what a poet should look like, and compared that mental image with the person before you).

They would never do such a horrible thing (if they had the opportunity; unless they absolutely had to). 

He wouldn't harm a fly (unless he absolutely needed to, if he happened to be beside a fly that was annoying him).

(Unless means if...not, so it counts as an if).

And where would that be, pray tell? (if such a place even exists - I  am so skeptical of your statement, that I  doubt the very existence of the place you mentioned).

The if is always present but often hidden. Do not use the conditional would unless you are sure it is in there somewhere.

So, when you come to write would (or could), ask yourself, "Where's the (hidden)  if?"