Rulings
Q1: Does this card's effect also activate when an opponent's Digimon with the Blue Flare or Twilight trait is played during my turn?
Yes, it does.
Q2: During my turn, a card with the Blue Flare trait was played and this card's effect was triggered. If a card with the Twilight trait is played by another effect before this card's effect activates, can I perform both Draw 1 and "return 1 Digimon card with DigiXros requirements from your trash to your hand" in this card's effect?
No, you can't. If Digimon with the Blue Flare and Twilight traits are played by separate effects, this card's effect will separately trigger for each trait. Because of the rule that states that an effect triggered afterward will take precedence for activation, for the situation stated in your question, ""return 1 Digimon card with DigiXros requirements from your trash to your hand"" will activate by playing the Digimon with the Twilight trait, but because the effect is a Once Per Turn effect, Draw 1 doesn't activate by playing the Digimon with the Blue Flare trait.
Q3: This card has the Blue Flare trait, so do I get Draw 1 from this card's own effect when this card is played?
Yes, that's correct.
Q4: Can I perform both Draw 1 and "return 1 Digimon card with DigiXros requirements from your trash to your hand" in this card's effect when a card with both the Blue Flare and Twilight traits is played during my turn?
Yes, you can perform both.
Data Sources
Lv.3 Gaossmon EX4-014 C Rookie | Virus | Reptile/Blue Flare [Your Turn] [Once Per Turn] When a card with the [Blue Flare] trait is played, <Draw 1>. (Draw 1 card from your deck.) When a card with the [Twilight] trait is played, return 1 Digimon card with DigiXros requirements from your trash to your hand.
Lv.3 Gaossmon EX4-014 C Rookie | Virus | Reptile/Blue Flare [Your Turn] [Once Per Turn] When a card with the [Blue Flare] trait is played, <Draw 1>. (Draw 1 card from your deck.) When a card with the [Twilight] trait is played, return 1 Digimon card with DigiXros requirements from your trash to your hand.
Lv.3 Gaossmon EX4-014 C Rookie | Virus | Reptile/Blue Flare [Your Turn] [Once Per Turn] When a card with the [Blue Flare] trait is played, <Draw 1>. When a card with the [Twilight] trait is played, return 1 Digimon card with DigiXros requirements from your trash to your hand.
Lv.3 ガオスモン EX4-014 C 成長期 | ウィルス種 | 爬虫類型/ブルーフレア 【自分のターン】［ターンに1回］特徴に「ブルーフレア」を持つカードが登場したとき、≪1ドロー≫（自分のデッキからカードを1枚引く）。特徴に「トワイライト」を持つカードが登場したとき、自分のトラッシュから、デジクロス条件を持つデジモンカード1枚を手札に戻す。