Do you have a cable tester just to check connectivity?
It depends, sometimes I won't check, because it's rare for something to go wrong if the tools are already reliable, and you can visually inspect, and then just try it for real. Besides, the simple cable testers only check for continuity anyway, which you could do with a multimeter (just poking the test probes gently against the visible contacts) if you've got both ends of the cable nearby. I have a slightly better cable tester (something called
PockEthernet) but that's excessive unless you're in a career where you're working more intensively with the cables.
Specialized insulation strippers, or just cut the sheath with a knife and use my Knipex on the wires?
Actually you just need to strip the sheath, and any craft knife can do that. The insulation on the wires is to remain (the metal contacts dig through the insulation).
What's good cable and connectors?
Any cable should be fine, provided it's advertized at supporting your needs. Cat5e ought to be sufficient (surprising to want to install Cat6 unless you're a network installer in which case you'll have better tools etc). Personally, I would try to get decent connectors though. There are huge tubs of very cheap connectors on Amazon etc., but I worry about the terminals being oxidized (I'm no metals expert though; I just want to eliminate the risk, and for the occasional cable, the cost of connectors from a reputable supplier is in the noise, if you don't need a lot). You'll want to check that the cable and connectors are both either for solid core or both stranded (i.e. the connectors are different for the two wire types). Also, for some luxury, you can get the soft plastic covers so that the lever on the plugs doesn't snag.
I just once in a blue moon need a cable for something and it would be nice to just measure out cable in situ, then cut it and add connectors. Easier to run through walls this way, too.
Ordinarily through walls you'll be terminating to sockets, in which case you won't be needing the RJ45 crimp tool. But occasionally people do just use the plugs on both ends, it's not very orthodox, but whatever you feel comfortable with.
Also, despite having the tools, and cable, I still sometimes just buy the patch cables. They can be low-cost enough to always keep a few spare. But sometimes I will make them, and of course sometimes a non-standard cable might be needed.