If the Gemini monster is face-down, it is flipped face-up, you inflict damage, and then the Gemini monster's effect applies and it is treated as a Normal Monster. Because of this, it counts as an Effect Monster, and contributes to the damage dealt.
A Gemini monster does not leave the field when you Summon it again. All Equip Cards will remain equipped to the Gemini monster after it is summoned to be treated as an Effect Monster. Effects such as "Premature Burial"'s destruction effect will still apply if "Premature Burial" is destroyed, even if the monster has been summoned to be treated as an Effect Monster.
When a high-level Gemini monster is Summoned again, you do not have to offer Tributes again for the Normal Summon.
Normal Summoning a Gemini monster to have it treated as an Effect Monster is not a different type of Summon rule. It is an act where you Normal Summon the monster once again when it's face-up on the field.
Since it is a Normal Summon, you can activate cards like "Trap Hole" when a Gemini monster is Summoned again.
If a Gemini monster activates an effect (like "Doom Shaman" activating his effect to Special Summon a monster) and the Gemini monster is removed from the field or flipped face-down, then the effect is not negated (unless the effect specifically requires the Gemini monster to remain face-up on the field).
When an Effect Gemini monster is flipped face-down, it is treated as a Normal Monster when it is flipped face-up. You must Gemini Summon it again to get the effect.
These TCG rulings were issued by Upper Deck Entertainment and have since been deemed unofficial by Konami. They were previously considered official, but may or may not necessarily hold true nowadays.
"Pulling the Rug" cannot be used to negate the second Normal Summon of a Gemini Monster, since the it does not begin a Chain.
Mentions in Other Rulings
Black Garden: If you have a Gemini Monster on the field and you Normal Summon it to change it into an Effect Monster, "Black Garden's" first effect activates. It will halve the Gemini Monster's ATK and Special Summon a "Rose Token" onto your opponent's side of the field. If the Gemini Monster's ATK had already been halved by "Black Garden" when it was previously Normal or Special Summoned, it is halved again.
↑According to Konami's new Fast Effect Timing rulings for the TCG a player cannot activate the Ignition Effect of an Effect Monster after the successful Summon of their monster, which makes that Ruling invalid.