Will the miles needed for upgrade depend on the ticket class?
 

If I was able to purchase a heavily discounted fare, will the required miles for upgrade to the next seat class still be based on their upgrade chart? Will there be restrictions on the ticket prohibiting upgrade? Or will I be required to use more miles for the upgrade?

The miles for upgrade are always the same. HOWEVER - you need to have purchased a fare class that allows upgrade. These are:

Economy: Y B H K M
Premium: W R
Business: J C D I (all)

Heavily discounted fares are usually not in these fare classes (unless you're looking at Business class).

Award charts and T&C are here: http://www.asiamiles.com/am/en/redeem/charts#2

So if I bought a ticket not in the allowed fare class, there's absolutely no way to do an upgrade with miles?

We are going to book a dozen tickets through a travel agent with huge discounts, and I was thinking to upgrade using my miles.

By the way, can I upgrade only the longer flights in a multi-flight itinerary?

a travel agent can book you into the appropriate fare class eligible for a one class upgrade. typically the savings you get from an agent are for fare classes where you won't earn any miles or club points. you can attempt to upgrade as many legs as you wish.

Correct.