It does appear that the space is too small for an implant and opening space would be challenging. It is probably easier to close the space with orthodontic treatment. We commonly do this for patients that have missing teeth. Once the space is closed it would be difficult for anyone to notice that you are missing a tooth. Basically your first pre-molar will take the place of the missing canine. We can even do a little reshaping of the premolar to have it look and function more like a canine.
When a patient comes into our orthodontic office they are paying for a service. Your orthodontist is going to give you the result while utilizing the tools necessary to get that result. If the doctor can give you the result with a clear tray from company A or B shouldn't matter, both companies make trays that exert a force on the tooth to get it to the desired position. As for treatment time, the process of moving a tooth through bone takes time. Generally, the farther a tooth has to move the longer it takes. Orthodontics accomplishes this movement by putting a force on the tooth to get it to move. If there is no force then there is no movement. An orthodontist uses brackets (lingual or traditional) with wires, trays (invisalign or clear correct) , appliances, and sometimes retainers with springs to place the ideal force to move a tooth. Your orthodontist can reduce treatment time by making sure you do not have any months where there is NO force on the tooth. Visits to the office are generally to make sure the teeth have the correct force and direction. If one method was better at moving teeth and reduced treatment time, then most likely other methods would be considered outdated and not used anymore.
A crossbite is when the width of the upper arch is narrower than the bottom arch. An expander is a device that widens the upper arch and is generally used when the crossbite in the upper arch is moderate to severe. Orthodontics alone whether it is invisalign or brackets with a wire are pretty good correcting mild crossbites however may not be successful in correcting severe crossbites. So depending on your crossbite you may need to add an expander to get the desired result.
Orthodontic treatment to correct a tooth positions can be done using a variety of treatment methods. Incognito and Damon are brands of braces. Incognito is a brand of lingual braces and Damon is a brand of traditional braces. Any orthodontist should be able to use their preferred treatment method and get a great result. As far as the bottom teeth. I suggest you see how your teeth look when they are straight and you might see the proportions of you bottom teeth may look ideal. You can always remove length but it is difficult to add length. So take a look and decide that when everything is staightened.
It is best to have your orthodontic work fully completed or near completion. Often times, restorations including veneers will turn out much better if you have straight teeth. I agree with the dentist get temporary bondings to fills in gaps while you are in orthodontic treatment. Also, we can definitely reshape your teeth. We like to also try to do this at the end of ortho treatment when the teeth are straight. Your orthodontist will be able to determine if in fact your teeth are proportionately longer . Usually teeth appear long because they are just hanging lower but when the orthodontist levels the teeth through orthodontics the tooth that the patient thought appeared long was in fact normal length.