In my experience, rice quality has a pretty substantial amount of variation, but the cooking device has little to do with it.
While I'm sure that most rice cookers on the market from Japanese firms are optimized and tested for short-grain, japonica rice, I've never had issues cooking basmati or jasmine rice in one, and I've even used them for farro and mixed grains.
I doubt that the rice cooker has much to do with it. I've made rice in heavy cast-iron enamelware on the stove, in a cheap Southeast Asian electric steamer without any fancy "fuzzy logic" electronics, in a fuzzy logic rice cooker, and an mid-range IH rice cooker, which is what we use at home now, and the quality of the rice and its age has a greater impact than the cooking method. I would say that the IH rice cooker produces superior results over our old fuzzy logic cooker, but it's certainly not an order-of-magnitude kind of difference.
Assuming you're located in the US, my benchmark go-to rice brand is "Tamaki Gold", which is from a japonica strain called koshihikari, and I think most of this brand's rice is grown near Sacramento, CA. It's more expensive than the typical Botan or Niko Niko brand calrose rice that's ubiquitious in Japanese supermarkets, but I find the quality far superior, and it's still a good value. My wife tends not to appreciate the Niko Niko or Botan calrose rice very much at all, so we don't eat it at home, but probably 80% of Japanese restaurants in the US are using it or a similar product.
Generally, I'd recommend staying away from the absolute cheapest brands, and choose something that's a couple of notches above. We've used imported rice from Akita or wherever and gotten very nice results, but the differences were far subtler than the price (on the order of $10 vs. $35 for a similar quantity). The sweet spot for quality is near the median price, assuming you're in a shop that offers a wide variety of options.
Edit: Sorry, until your edit, I didn't realize you were working from parboiled or converted rice, I assumed raw rice, as I've never heard of anyone cooking the quick cooking rice products in a rice cooker. (Rice cookers often have their own "quick mode" which shortens cooking time with normal raw rice at the cost of a slightly reduced textural quality). Considering that's what you were starting from, consider using other supermarket brands of rice that meet your target grain size and stickiness/fluffiness, but aren't marketed for speed or convenience.
Because we don't do recipe requests around these parts, I'm going to treat this as an ingredient question.
Japanese markets (and some online merchants, including Amazon) often sell dried versions of the seaweed mix that many restaurants use. It'll be sold as kaiso salad or seaweed salad. Some versions are better than others; some contain a ton of dyes that make the salad look unnaturally neon to my eyes, although the taste and texture is similar. Such blends typically contain wakame and other seaweeds, including agar agar noodles, for example.
If not included in the package, you can make a simple dressing based on toasted sesame oil and Japanese vinegar, in roughly a 1:3 ratio, or use sesame oil and lemon juice (maybe 1:1-1:2 ratio since it's more acidic). You may want to include soy sauce or salt. This is loosely in the category of sunomono, so it doesn't typically contain emulsifiers like mustard, but if you find it hard to blend you can use a touch of mustard or something like lecithin or a stabilizer like xanthan gum.
Best Answer
I've eaten a few of these before and while it sounds like furikake to someone who hasn't had them, I don't think that's what you're looking for in this case. Furikake is primarily used for seasoning white rice to be eaten otherwise plain, but I'm almost sure you're looking for something like this:
(from http://jpninfo.com/22660)
I don't think there's a special Japanese name for these seasoning packets and the front just says "Shrimp Fried Rice", "Salmon Fried Rice", "Crab-flavored Fried Rice" and so forth. I would call it a seasoning packet, much like I would for taco seasoning. I wouldn't be surprised to hear a Japanese person call them paketto or something similar loanword-y.