As far as a torque/contra spring goes, since you are already going to order from racing planet, they can be found on this page:
http://www.racingplanetusa.com/transmission-parts-c-2_26378_26382-1.html?sessID=8daad231bd2b7945f800b06bc850e5adWith a Jasil/Top Racing 15% spring (comparable to other brands' 1000rpm) and 8 gram roller weights, my top speed is currently 41-44mph, dependent on weather conditions. The main difference with a spring (that I have found) is ability to maintain speed when encountering wind (10-20mph wind drops my speed to 32-35mph versus 25-28mph with the stock spring. Note:40mph wind gusts and all bets are off - I went from 35mph to 25mph when turning into that speed of wind gust).
At present, I have no experience with sliders, so cannot say one way or the other.
With your set-up as it currently is, an 82 main jet will benefit, if it hasn't already been done. With the new pipe, an 85-88 will do well for the scoot. I have become a fierce proponent for the plug chop! Get some extras! (I don't know your location, but if in the U.S., partsforscooters
http://www.partsforscooters.com/145-31_NGK_Spark_Plughas the best prices and quickest turnaround.)
For tools, I use a torque wrench & a 3/8" drive 12"breaker bar and sockets to access variator and clutch nuts. A clutch-holding tool works well for the clutch (supplemented by using the parking brake) - I purchased mine at treatland
https://www.treatland.tv/universal-holder-tool-p/hold-anything-super-tool.htm since they had the best price, but any purveyor you choose, so long as it is like the tool listed at the link, will suffice.
There are many ways to hold the variator. I got a holding/blocking tool from racingplanet
http://www.racingplanetusa.com/variator-holder-blocking-tool-kymco-horizontal-p-256-1.html?cPath=2_26378_26382&sessID=8daad231bd2b7945f800b06bc850e5ad and while it isn't an exact fit for our case, it can be used quite effectively once figured out.
As to your stalling at idle, I would check the idle speed screw on the carb. Prior to starting, you can figure out how far it is currently out by counting the number of turns it takes you to screw it all the way in and then back it out the same number of turns. With that knowledge, start the scoot and let it warm up for 10-15 minutes by taking it out on a short run (this allows the auto choke to do it's thing so it won't interfere with your setting the idle.) After the run leave the scooter on, put the scoot on its' center stand, and adjust the idle screw. It may be that it is too far out. Mine is currently about 2.5 turns out, but I will likely take it in to 2.25 turns.
I am unfamiliar with the Dr. P vari, but anything is better than the stock one. Just make sure it will fit.
As to carb replacement, I am still researching this, myself. 17.5 will see an increase in performance and adjustability, but if a bbk is anywhere in your future, a 19 would be best.
The Like is a great little beast, with some great lines to it.
Welcome to the forum!