Rice noodles are perfectly allowable on the Perfect Health Diet, which falls under the Paleo spectrum and quite a few of us follow, white rice being among the "safe starches." But if you're just starting with Paleo it is best if you stop looking for "versions" of the bad stuff you ate before and focus on whole foods.
Most everybody would consider even gluten-free noodles to be a cheat, to be avoided. Though most will probably agree that a gluten-free noodle is a pretty mild cheat. Rice noodles are probably the best of the bunch, as rice typically causes few people issues. Corn is also fairly well tolerated by most people.
But again, if you're just getting started with paleo, just avoid noodles altogether.
On "Paleo," no, there are no noodles allowed. You can spiral vegetables, or use spaghetti squash, or there is a recipe floating around for pork skin noodles.
Make ribbons with a vegetable peeler on a zucchini and boil for 1 minute. Top with sauce. Spaghetti squash is also great and makes enough for a few meals for my family.
If you want something more "real", I would go for the rice noodles. But I wouldn't make them a staple. Maybe once or twice a month. I haven't tried the kelp noodles, but they sound promising.
It's generally best to avoid noodles of all kinds especially if you're new to eating this way. Eating substitute foods just keeps the cravings for the food alive, and because the substitutes never taste or feel quite right, you're more likely to revert to the real (really bad for you) thing.
While ITA with Karen and PrairieProf about breaking associations, if you do want noodles, you could try the kelp noodles that are available. Not only are they "paleo," but they're strict Whole 30-compliant!
I also will sometimes eat Eden Foods buckwheat soba noodles: be warned, though, that there are multiple versions and you have to read labels. The standard ones are a mix of buckwheat and regular wheat. You need to get the 100% buckwheat ones to make sure they're gluten-free.
ETA: despite their name, egg noodles are not just egg. They've got plain old white flour in them! (I remember my grandmother and my dad making them for soup when I was growing up.)
If do not have a problem with rice, than rice noodles can be fine, just make sure you read the ingredients.
That being said, I agree that eating dishes with noodles should not be part of your daily diet, and you should stick to eating foods that are minimally processed.
Kelp noodles! They are made by a few companies: Organic Lives and one other. Google them. I also had "noodles" made of shredded coconut meat the other day which were truly amazing! Give it a shot.
When I do pasta, I do Andean Dream quinoa pasta. It's not 100% (is there such a thing??), but it's entirely quinoa (first ingredient) and brown rice. Two grains I know I can tolerate well. Win!