Yes, the Chapin Tree/Turf Pro 4 Gallon BackPack Sprayer (#62000) can be used with both insecticides and herbicides. Be sure to triple rinse and thoroughly clean your sprayer in between products though.
The Chapin Tree/Turf Pro 4 Gallon BackPack Sprayer and the Chapin Slide Pump Sprayer are not used together. The Chapin Slide Pump will spray over 30 ft alone by mixing your solution in a container and dropping the weighted strainer in the container.
The Chapin Tree/Turf Pro 4 Gallon BackPack Sprayer (#62000) requires that you pump continuously while you spray. This type of sprayer will not hold pressure like regular hand held pump sprayers. You do not have to take the sprayer off to pump it.
Yes, you can use a sprayer such as the Chapin Tree/Turf Pro 4 Gallon BackPack Sprayer (#62000) with wettable powder formulations though you will need to agitate the solution so that the powder does not settle at the bottom and clog the sprayer.