There are a number of soft hyaluronic acid fillers that can be used in the lips. In fact, several new products have been introduced to the market since the time this question was posted. Several factors might be used to determine which specific filler is the best choice for a given patient, such as cost, amount of plumping desired (vs smoothing), duration of effect desired, and other areas that might be treated in conjunction. Probably more important than the product itself is the injector's technique and aesthetic eye. I recommend that individuals who are interested in lip augmentation first obtain a consultation with an experienced injector and ask to see their before and after results, etc, before deciding to proceed. Good luck!
Sometimes you will see a subtle 'tug-of-war" effect, where relaxing one set of muscles will make the action of the opposing muscles relatively stronger. In the case of treating the "11's", the effect is actually a medial brow lift if the frontalis muscle (responsible for forehead lines) is left alone or weakened to a lesser extent. This is usually considered a positive effect, but the horizontal forehead lines might appear subtly more apparent because their action is unopposed. Also, if you relax the wrinkles in one area, it can make the wrinkles in other areas become more noticeable in contrast. If you have minimal wrinkles on your forehead, you should be fine just treating the corrugators alone. Many patients start with that area and are very happy with the results. Also, It's easy to go back and treat the horizontal forehead lines with a small dose to rebalance the muscles if you feel it had any negative effect on the forehead.
It seems like you are concerned about upper eyelid hooding. There is a sort of tug-of-war between the muscles that lift the eyelid and the muscles that depress it. Botox relaxes and temporarily weakens the effect of the muscle. Depending on where and how much the botox is placed effects the tug-of-war, and will either contribute to lifting or depressing the eye lid. In general, injecting the outer frontalis muscle depresses the lateral lid, while injecting the muscles that create the crow's feet (obicularis oculi) muscles will open up the eye a bit.
Many clinicians and patients seem to like this exaggerated lift, but I agree with you that it tends to look unnatural. It might look fine at rest, but when the brow is raised, it looks "Spock"-like, and is a tell-tale sign that you have had botox. Generally this can be avoided by strategic placement of Botox. 20 units is a lot for the frontalis muscle in my opinion, and in your case, it seems to have been concentrated in the middle 2/3 of the forehead. At this point, you have 2 options. One, you can have your clinician inject more in the lateral aspect of the forehead, particularly lower, near the peak of the brow, but this will have the effect of dropping your lateral brow slightly at rest, so small increments are best. The other option is to wait it out. The results of botox usually do soften a little after 3-4 weeks, so it might not be as noticeable in a few weeks. To avoid this in the future, I would recommend fewer units and to place the units in a way that the effect tapers more gradually to the sides.
A heavy brow after Botox injection of the forehead is, unfortunately, a common complication if muscle balance and anatomical variations aren't considered. Injection of Botox in the forehead (frontalis) muscle is designed to smooth the horizontal forehead lines, not lift the brow. Unfortunately, many people with prominent horizontal forehead lines have a heavier brow to start with... and are very sensitive to weakening of the frontalis muscle. Sometimes we can create the appearance of a lateral brow lift by relaxing the middle of the forehead more than the sides, but the only real lifting of of the brow with Botox comes from treatment of muscles between the brows and treatment of the upper crow's feet muscles. Without photos, it's hard to know exactly what you are experiencing. The good news is that even though Botox usually lasts 3 - 4 months to wear off completely, most people feel it's "heaviest" in the first few weeks after taking effect, and then eases up a bit.