Do jumbo and smaller airliners' horizontal stabilizers provide any lift? Apart from maintaining the aircraft's angle of attack, can they perform the dual function that includes adding some lift as well? Are there any aircraft designs that use horizontal stabilizers for lift as well, apart from delta wing aircraft?
The horizontal stabilizer provides lift, but usually in the negative direction. Could one provide positive lift?
The answer depends on cg location.
Forward CG, the answer is no.
Aft CG, the answer is yes but only if the CG was aft of the center of lift.
In other words, the more forward the CG is compared to the center of lift, the more downforce the tail must provide. The lift from the horizontal stab in the "wrong direction" must be countered by the wing, and thus there is more induced drag. For a given aircraft gross weight and thrust/power, there will be a higher cruise speed with a more aft CG.
There are practical limits to extreme fore and aft CG positions, and so each aircraft has an allowable "envelope" for loading for safety margin.
Aircraft are designed to have a range of allowable CG locations to make loading easier before the flight (especially for cargo aircraft), and also to allow for passengers & crew to move about the cabin in flight.
Horizontal stabilizers could be sized smaller but that would decrease loading flexibility, decrease passenger comfort, and decrease aircraft handling performance; often all airfoils and control surfaces are designed with many factors in mind, for good control harmony, good ride in rough air, for stability on instrument approaches, for flexible loadings of various belly and main deck cargo holds / passenger configurations, etc.
Flying wings and delta wing designs come to mind as aircraft without horizontal stabilizers, such as the Northrup Spirit or Convair Delta Dart.
Your other question, would manufacturers design horizontal stabilizers to provide lift? Hopefully you can see by now this is not a design goal. Wing design is so advanced that designers are making amazingly beautiful and high-performance wings that are most efficient at creating lift. Look at a Gulfstream V wing or Boeing 747 wing. It is not efficient to create a wing that doesn't work great and then needs an additional surface to help make more lift. Exceptions to that might be bi-planes or tri-planes, as they used several shorter wings instead of 1 longer wing, but that is not practical in air transport. How many Boeing Biplanes and Airbus Triplanes do you see traveling the globe? It's simply better to put all the lift needed into a single wing design.
As a general rule, if the small surface is forward of the main wing, it lifts up. If it's aft of the main wing, it lifts down (most of the time).
This is a gross oversimplification, but it's very useful to help with imagining what is going on in your mind.
An airplane with a normal tail is like a seesaw, except the pivot is not a solid point, but the middle of the lifting force generated by the wing. One end of the seesaw has the center of gravity trying to tip it forward, like a kid sitting on the seat, and the other end is you pushing down to hold the kid up, or in the airplane's case, the horizontal tail pushing down against the moment of the center of gravity acting forward of the center of lift, to keep it in balance. If you broke off the horizontal tail, the airplane would point straight at the ground, as if you let go of the seesaw seat on your end and let the kid drop.
An airplane with a canard surface at the front and a large wing at the back is more like a dump truck. There is a big axle at the back holding most of the weight, and a little axle at the front holding up a minority of the weight.
Both axles are holding up - and both the canard airplane's surfaces are lifting, but the ones at the back are doing most of the actual work. The center of gravity is close to the end with big axle (the main wing), and the axle at the front end (the canard surface) only doing a small portion of the total holding up (or lifting) because its job is mostly about control (control over pitch and speed in the airplane, control over steering and braking in the truck).
Airplanes with three surfaces, both a tail and the back and a canard on the front, are kind of like a seesaw but with a little support stick under the kid to help hold him up so you don't have to push down as hard at your end (the tail). Kind of a seesaw/dump truck hybrid you might say.
Any canard design will use the horizontal stabilizer for lift.
Airliners historically used downforce on their tails in order to have sufficient static longitudinal stability, but the more recent designs use a small amount of lift in cruise and keep stability up by artificial means. A tail tank is sometimes used to shift the center of gravity sufficiently back.
Tail lift in trimmed flight varies when a cambered airfoil is used. Wings using cambered airfoils will usually result in positive tail lift at high angle of attack and moderate to rear center of gravity locations. Usually, the loading of the tail should be low in order to allow for enough margin for maneuvering.
Note that the negative airfoil camber at the root of a tailplane helps to keep the isobars on the sweptback surface parallel up to the root, so this is not immediately an indication of predominantly negative tail loads. Negative nose camber on the tail also helps to delay flow separation on the tail at high negative loads, which are needed to trim the aircraft with full flaps.
The opposite. Generally the tail pushes down on the aircraft. While the wings push the fuselage up, the horizontal stabilizer pushes the tail down. Yes, this does increase load on the wings.
Look closely at the airfoil on a horizontal stabilizer, you can see it is inverted.
Also, if you look at the range of motion on fully movable horizontal stabilizers on jetliners, the vast majority of their range is negative AoA.
For stability reasons tails generate a down force.
It would in theory be possible to have a horizontal stab generate lift (upward force) by compensating for the resulting instability with a fly by wire system, but the wing/tail system would always be less efficient at generating lift then just making the main wing a little bigger.