For tightline nymphing, or commonly referred to as Euro nymphing, you are going to want to use a longer and lighter rod in order to reach out and high stick the water you are fishing.
My favorite rod that I use is the Thomas & Thomas Contact 10’8” 3 wt. The sensitivity in the rod helps detect strikes and the bottom, and the length allows you to fish out away from you further.
As far as slow or faster action rods go is really up to personal preference. I like a medium-fast action in the spring and fall when the water flows are low, but in the summer when the water is 4 times higher, I will go to a faster action rod to combat the higher water.
Since 30 feet out is about the maximum you can effectively fish when Tightline Nymphing, a couple of extra inches of rod can certainly help.
I describe my leader system, how to build it, and a lot more information in my Tightline Nymphing article.
The Scientific (SA) Competition Nymph Line or the Rio Euro Nymph line are what I use. It is .022” in diameter this allows for less “bow” in the line when high sticking. Since flies are what we use to load the rod instead of the line loading the rod normally and the heavy leader turns over the rig, the fly line can be light to decrease drag in the air and water.
Some people like to use their euro rod as a hybrid. You can put a 3-4 weight line on it of your choice and switch out leaders when you want to dry fly fish or put a small wool indicator on and use it as a indicator rod. Since the leaders are usually about 20 feet in length the line bowing isn’t a problem if you are fishing within the parameters.
I personally stick to the Competition Nymphing line when Tightline Nymphing and have an indicator rod on stand-by if I decide I want to do that or if it gets windy. You can also just stick with 50-60 feet of straight 20-30lb mono line. The problem with this though is that it coils easily especially if it is not used consistently.