In the film universe of Alien (1979) and Aliens (1986), are the ships light speed capable?

Yes, the ships are capable of speeds significantly above the speed of light. From in-movie clues this is fairly clear, e.g. from Alien dialog:

DALLAS: How far to Earth.

LAMBERT: Ten months.

Then later in the meeting they have:

DALLAS: Well, some of you may have figured out we're not home yet. We're only halfway there.

So the journey they are making is expected to take 20 months. A journey between stars below light speed would be years, decades or even longer. This isn't their 'local time' due to some relativistic effect of going at high sub light speeds, as Ellen Ripley is shocked when she finds that she slept for over 50 years between Alien and Aliens, and that her daughter had grown old and died.

In Aliens the Sulaco is dispatched LV-426 to investigate why the colonists had stopped communicating. Its clear its its to investigate and potentially rescue them ("its a bug hunt") and they find cocooned colonists, so its clear this is a fairly fast military ship.

I don't know where the information comes from, but the Alien v Preditor Wiki states that the speed of the Nostromo is 0.12 light years per day with its cargo.

As for the issue of automated ships being superior for such long voyages - this might be true, except that people are more adaptable and creative problem solvers - and putting your trust in people when you are ferrying huge quantities of (presumably expensive) mineral ore might be seen as a good investment, even if they require more infrastructure to keep them alive than automation or androids.

Yes, either the ships can travel at theoretical speeds beyond that of light or they can enter and exit "warp" or "hyperspace" corridors which allow travel across vast areas of space in far less than time than traveling at or near light speed would. Some clues from the 19780 novelization of the film Alien and the subsequent Alien films are:

1. In the novelization, the Nostromo's cargo is a refinery of petroleum which is still used for plastic on a future Earth. It isn't conceivable that an existing trade route would be established when cargo deliveries took centuries or millennia for delivery.
2. Carter Burke makes the comment during Ripley's dream that her escape shuttle drifted outside of the space lanes, implying that there are normal travel routes through space in the Alien universe. This again would require FTL (faster than light) travel to even be possible.
3. Burke tells Ripley that contact was lost the colonists on LV-426. Since normal communications with another planet would take years,decades or even centuries, it can be inferred that faster than light communication is possible. To believe in that, then faster than light travel also has to be possible.
4. When Ripley asks Hicks how long it will be before they can expect rescue if they aren't heard me from he replies "Seventeen days." Given that the events of the film do not occur within the Sol system of planets, it can be inferred that some form of FTL is possible to make the journey in such a short period of time.
5. In all of the Alien franchises, sentient alien life is not viewed to be unusual. Given the fact that no such life exists in the Sol system of planets, one can infer that that series occurs outside of this area. That would only be possible if FTL was/is possible in their universe.

Absolutely they go faster than light......I've been a big fan of the Alien films for 40 years. I remember reading the novelization in middle school......and I remember a part where Ripley describes the acceleration of the Nostromo and the effects when passing the speed of light. I think she said something to the affect of how the stars in front would turn blue and the stars in the rear would turn red... then a flash of light all around the ship....... Something that can be seen in the movie The Last Starfighter when the starcar first takes Alex Rogan off world.