1) When PSY-Frame monsters are summoned multiple times during the same chain, PSY-Frame Circuit triggers more than once, but the official rulings state that it can only be activated once in this case.
2) It shouldn't be possible to activate a Bottomless Trap Hole after a Gear activates its effect, since the last thing to happen is not the summon of the Gear and Driver (with Alpha the last effect is the search, with Beta is the part that ends the Battle Phase, with the others is the destruction of the negated card). This has been stated multiple times by judges at YCS Dallas, too.
3) Sometimes PSY-Frame Lord Omega's effect that returns banished cards to the Graveyard allows to see the opponent's cards that have been banished face-down.
4) YGOpro allows atm to activate both Alpha and Gamma when a monster is summoned and its effect is triggered (it should be possible to activate only one). This happens with other cards as well, like summoning Manju, destroy it via Ghost Ogre & Snow Rabbit as CL2, then activate Kagetokage as CL3. I'm guessing this means there's nothing wrong with the individual scripts, but with how the rules are currently enforced by YGOpro
5) Not really a bug, but a few extra SetHintTiming for the Lords would make PSY-Frames much easier to play:
- PSYFrame Lord Zeta: every time a monster is summoned during the opponent's turn (including Zeta itself)
- PSYFrame Lord Omega: every time a monster is summoned during the opponent's turn (including Omega itself) and at the end of each of the opponent's Main Phases

Every bug I found happened with either YGOPro Android 1.4.7 or YGOpro 1.033.6 (which should have the same scripts).

I wrote directly to Konami about a few things that were unclear:

1) Let's say that the opponent Normal Summons a Manju and activates its effect while I have PSY-Framegear Alpha and Gamma in my hand. In this case, can I activate only one of them or both, and if it's the latter case in which order?
2) If I activate PSY-Framegear Gamma, Delta or Epsilon as Chain Link 2, will my opponent be able to activate Bottomless Trap Hole or Torrential Tribute after that chain has resolved?

This is what they answered:

1. You can only activate one or the other, not both. Alpha has a “When… you can…” trigger effect so you have to activate it as soon as it’s your turn to activate a card or effect. Gamma on the other hand is a Quick Effect that must be Chained directly to the monster effect you want to negate. As a result, both cards have to activate at the same time, so you have to choose which one you want.

2. Yes. These cards don’t negate the effect of the card they’re Chained to, they negate the activation of the effect they’re Chained to, eliminating that Chain Link altogether and making the Summons the last thing to happen.

This means that the interaction of the gears with BTH is fine as it is, but the 4) bug I reported has been confirmed.

I wrote directly to Konami about a few things that were unclear:

1) Let's say that the opponent Normal Summons a Manju and activates its effect while I have PSY-Framegear Alpha and Gamma in my hand. In this case, can I activate only one of them or both, and if it's the latter case in which order?
2) If I activate PSY-Framegear Gamma, Delta or Epsilon as Chain Link 2, will my opponent be able to activate Bottomless Trap Hole or Torrential Tribute after that chain has resolved?

This is what they answered:

1. You can only activate one or the other, not both. Alpha has a “When… you can…” trigger effect so you have to activate it as soon as it’s your turn to activate a card or effect. Gamma on the other hand is a Quick Effect that must be Chained directly to the monster effect you want to negate. As a result, both cards have to activate at the same time, so you have to choose which one you want.

2. Yes. These cards don’t negate the effect of the card they’re Chained to, they negate the activation of the effect they’re Chained to, eliminating that Chain Link altogether and making the Summons the last thing to happen.

This means that the interaction of the gears with BTH is fine as it is, but the 4) bug I reported has been confirmed.

Sorry this took a while the answer you got for Alpha+Gamma did not seem correct so I asked the OCG Konami and they gave this answer:

Q.
サイフレームギア・アルファ + サイフレームギア・ガンマ

私はこれらの私の手にあるカードと私の相手の召喚アルカナフォースモンスターを持っている場合、私はそれらの両方を活性化させることができますか？
A.
相手が「アルカナフォース０－THE FOOL」が『このカードが召喚・反転召喚・特殊召喚に成功した時、コイントスを１回行い以下の効果を得る』効果が効果を発動した時、チェーン2で「PSYフレームギア・γ」の効果を発動し、そのチェーン上でさらに「PSYフレームギア・α」の効果を発動する事ができます。

Q.
Sai frame gear alpha + Sai frame gear gamma

If I have a card and summons Arcana Force monster of my opponent in these my hand, I will be able to activate the both of them?
A.
Opponent "Arcana Force 0-THE FOOL" is "when this card is successfully Summoned or Reverse Summoned or Special Summoned, obtain the following effect is a coin toss once" when the effect was to trigger the effect, in the chain 2 "and to trigger the PSY frame effect of gear · γ ", you will be able to trigger the effect of further" PSY frame gear · α "on the chain.

Which branch of Konami did you contact to get your ruling?

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I contacted the North American office, so I suppose the rulings may be different.

EDIT: So, to summarize.
1) is the only one that is certainly a bug, since it's stated in the official OCG rulings.
2) is not a bug, the judges at YCS Dallas were wrong.
3) is actually not a bug, sometimes the face-down cards are shown face-up when Omega's 2nd effect is activated because the player knew what they were when they were banished.
4) may not be a bug (it depends if OCG or TCG rulings are followed).
5) was not a bug to begin with, but it still would be nice if something was done about it, although I suppose it doesn't have a high priority.

Up for Circuit's bug.