Cataract surgery is essentially lens replacement surgery. As such, by choosing the appropriate power of lens implant (based on calculations using parameters such as corneal curvature and eyeball length), focusing issues such as myopia and hyperopia can be 'corrected'. In fact, you can choose what kind of refractive state / spectacle power you would like after cataract surgery, whether that be emmetropia (distance ~0), or a small amount of myopia (~-1.50 to -2.00D).
Nowadays, there are also toric lens implants, which can help to reduce post-operative astigmatism by canceling out corneal astigmatism.
So it is quite straightforward to say that yes, myopia and astigmatism can be corrected with cataract surgery with or without prior LASIK. However, it is worth noting that prior LASIK can make the lens implant calculation a bit less predictable than for patients without such prior surgery.
Presbyopia is a bit different. There is actually nothing that really 'corrects' presbyopia. There are only different options with different pros and cons to reduce the need for reading glasses after cataract surgery.
Option 1, which is probably best for someone like you who has prior experience of monovision (either with LASIK or contact lenses), is to reproduce monovision with cataract surgery. That means you have cataract surgery, and use a monofocal lens implant to give good distance vision in your eye that previously had good distance vision, and use a monofocal lens implant to give good near vision in your eye that previously had good near vision. People with monovision sometimes still find it helpful to wear glasses, eg for driving at night. However, they do not need glasses most of the time.
Option 2, which is somewhat risky, is to consider multifocal lens implants. This option can potentially give good near and far vision without glasses, but there are quite significant potential cons. The most important of which include halos/starbursts around lights at night. Since LASIK sometimes can itself give rise to these visual disturbances, the addition of further halos by the multifocal lens can potentially lead to a situation that is difficult to tolerate. Multifocal lens implants in patients with prior LASIK therefore must be considered on a case by case basis, taking into consideration such factors as the prior spectacle degree before LASIK, quality of corneal topographic features, patient tolerance of visual disturbances, and whether the patient engages in night time activities such as driving a motor vehicle. It is worth noting that even without prior LASIK, some people just cannot tolerate the visual side effects of multifocal lens implants. There have been cases where multifocal lens implants had to be exchanged for monofocal lens implants because patients (without prior LASIK) could not tolerate the halos etc associated with multifocal lens implants.
So to recap, for someone who had prior LASIK monovision, and now has cataracts with myopia, astigmatism and presbyopia, probably having cataract surgery with toric monofocal implants aiming to recreate the prior monovision is the better option. This is the option requiring the least post-operative adjustment, since it is creating a situation which has been experienced before by the patient, and which the patient has already been used to for many years.
