I used the carb tablets for the first time last week. They are slow to dissolve, so don't rush the conditioning phase.
Don't rush the fermentation either. Let it go for the full three weeks. I don't do 'Mr. Beer', so I'm not sure how much sediment there will be. Shouldn't be much since you started out with syrup.
Don't fear the sediment in the bottle. Doesn't hurt a thing and you have other things to worry about. Clarity in brewing will come with experience.