Should I put a surge protector on the 110v to the Trane furnace

If you notice none of the described symptoms when you plug the laptop adapter directly into the wall outlet then you should consider discontinuing the use of this surge protector / plug strip. This seems to me to be a serious safety issue that needs to be addressed immediately.

Once you get back to a safe usage scenario you should do some checking of the laptop adapter for safety issues as well. Remove the adapter from the wall power and from the laptop. Now using a digital multimeter in the resistance (ohms) measuring mode check for any low resistance readings (less than 10 megohms level) between any of the AC power leads and any of the contacts at the laptop charger plug end. If you find any low resistance connections from the AC to DC side of the adapter then you should replace that as well.

(Note that all of the numerous laptop adapters I have encountered are supposed to be 100% isolated from input to output. I.e. NO input to output electrical connection).

Now back to your surge protector. Some surge protectors may have installed components such as MOVs or high voltage capacitors between the AC lines (hot and neutral) and the earth ground connection of the AC wiring coming into the protector. If they are wired this way it is possible for some people to feel a slight shock when touching any grounded metal parts of equipment that is plugged into the downwind power jacks of the surge protector. In my electrical engineering opinion I do not think this type of surge protection wiring is the best idea. Now at the same time some manufacturer of a surge protector may boast that their unit is superior (and thus way more costly) if it has MOVs attached line-neutral, line-earth, and neutral-earth. This can lead to low level leakage currents from the AC line to the earth ground connection. If your "high end" unit is like this you should consider not using it!

If your laptop adapter has a connection from its input AC leads to DC output side and it is used in conjunction with a surge protector that has multi-MOVs causing leakage currents to flow then this can lead to the slight shocks that you feel when touching metal parts of the laptop.

My recommendation - safety first when dealing with anything electrical attached to the AC power lines. At first indication that there is something amiss discontinue use immediately.

As explained in this Electronics.SE article, daisy chaining of AC power protectors, where they both attempt to protect the same electrical circuit, can cause interference and failure. But that's not what you're doing. You have 3 protectors, for electricity, ethernet, and phone/DSL respectively. Those are 3 separate circuits, even if some of them flow into the same devices, so your setup is fine.