Ryu-Oh kind of fulfills the same role, but Mordred is super good. The thing is, Ryu-Oh's Black Propaganda has a fairly bad hit-rate. With elemental advantage + Mordred's Debuff Resistance Down + 6% Affliction from Accessories, it STILL doesn't reach 100%. Close, but not guaranteed. So, if you're not going to run Mordred? It'll be even worse. You'll run Ryu-Oh without elemental advantage? Even wose. I doubt you have proper Accessories yet, so even worse. In other words - Ryu-Oh will miss A LOT if you intend to use her in every team. You might still want to since her A debuffs are super good as a new player, but don't rely on her.
tl;dr; No, Ryu-Oh doesn't replace Mordred.
R Hime are generally terrible, but Advent Ragnaroks are super weak. Once you have your Grid set-up, you'll blaze through these. Until then... either Mordred or Joan are your best bets. But if it's Dragon Eye Shards you're after, you should rather be looking for a strong Union. Top5 Unions get ~15 Shards a month. You won't make it one of those probably, but you could get into one that gets ~8 or more.
As I said in my thread, Hercules is super good. However, before you get to Hercules, get the other basics down. You have Gawain and Cass at least (btw Andromeda is a WAY bigger newbie trap than cheap Arthur...), but grab the rest of the basics and then Hercules. A lot of Hercules' power comes from her Relic weapon, and you're too weak to farm those at the moment. By the time you'll be able to get somewhere on those bosses and you can buy Relic weapons, then you're probably starting to be at the stage where you can drop Mordred. If you don't have Hercules unlocked at this point, you'll be sorry...To super simplify the damage formula, it's BaseAtk*CharacterAtk*ElementalAtk. If you play around in that, you'll notice through the power of mathematics that the "best one" is the "lower one." So if you have less CharacterAtk, then it's better to increase CharacterAtk.
The thing is, your weapon skills (Assault) are CharacterAtk. This is why your Grid is so important. You can reach 130% CharacterAtk(/Assault) just from your Grid. That means that the best thing to get from Eidolons (for damage) is ElementalAtk, since you already have up to 130% (10x slvl20 SR Assaults) from Grid.. This is why Elemental Eidolons are by far better for veteran players. For you? 40% CharacterAtk + 20% HP is probably way better than 45% ElementalAtk since I assume your Grids are weak.
Notice two things about ElementalAtk though - Relic weapons give 30% ElementalAtk and elemental advantage is 45% ElementalAtk. If you're running both (which you won't for a long time since you're not strong enough to farm Relics just yet), then you'll have 117%+ CharacterAtk and 75% ElementalAtk before your Eidolons. At this point, it's ideal to pick a P2W Eidolon from Friend List (100% Elemental ones) and bring your own CharacterAtk + HP. 157%+ CharacterAtk and 175% ElementalAtk, a good balance, plus you get 20% HP to boot.
But again, for now, your biggest limiter is your Grid. Farm SR/SSR Assault weapons from events and and Disaster raids, level those up with GemGacha.And it should be noted that Vohu is super good against Fire, but not all that good elsewhere.