Is it best to do a long set of lighter weights fast? Or is it best to do heavier weights and take your time?

Or should it be mixed up?

It all depands on your goals. If your looking to add muscle and size go for heavy weights and low reps, which if they are heavy enough for you, you shouldn't be able to go fast.
If your looking for muscular endurance you would go for lighter weights, longer sets and yes a little faster tempo.

Why not heavy and fast? Fast = as quick as you can while still maintaining control and form.

Why not HEAVY, FAST, and with a SLOW eccentric

What's your definition of a SLOW eccentric?

I'd say 'slower eccentric' than the concentric. 1 second concentric, 2 second eccentric.

just enough so you're resisting the eccentric, and not letting gravity do it for you.

By slow and heavy, do you mean they move slowly because they are heavy? If that the case then I wouldn't worry about that, what matters is how fast your body is trying to move them , not the actual speed they're moving at

If we are talking about training fast, both the eccentric and concentric are two slow in this example.

If training fast is the goal. TRAIN FAST. in all aspects of the lift.

If training slow is the goal. (The slow training has always made no sense to me) TRAIN SLOW.

If you want to get stronger. You have to lift heavier weights. To get heavier weights to move you have to try to lift them as fast as possible.

The intention of speed is what will make progress.

From my experience the amount of weight should be the indicator of bar speed. If it is lighter, you are moving faster. If it is heavier, you are trying to move fast, even if the weight itself does not move fast, you will have a greater result.

The higher the speed the more muscle is recruited. The more muscle that is recruited, the better results you will get.

Once your technique is good the faster eccentric phase will help to recruit even more muscle. The more efficient you are at taking advantage of the eccentric phase of the lift, the more weight you will be able to use.

If you are trying to have more muscular endurance or get very strong, the general advice is still the same.

1. Make sure you have good technique in the lifts that you are doing.

2. Move all your training weights fast. Focusing on the speed of the concentric phase of the movement.

3. Once your technique is dialed in, increase the speed of the eccentric movement as well.

To put it another way - You will start lifting in a controlled manner, once you have the basic technique of the lift down you will increase the concentric phase of the lift, once you have the technique down for that you can increase the eccentric speed.

On to your specific questions -

Sometimes. If your goal is muscular endurance this will be effective. Will this always be the best option, no, but speed of lift will be an important part of training.

Once you have exercise technique nailed down, the intention of speed under heavy weights is going to be very important.

If your training goals are general fitness, gain some strength and endurance, and look good. Yes mix things up.

If your goals are increased strength. Yes, mix things up.

If your goals are muscular endurance. Yes, mix things up.

Just remember. As your progress in your training career, increasing the speed of lifting will decrease the chance for plateaus and stagnation.