First off, it feels unnecessary and strange to me that you're pulling the Phyrexians into this design. Not only because the Phyrexian Invasion has been halted, but also because the Orzhov already deal with the afterlife, so pulling back souls for a price should be right up their alley anyway. Then there is the fact that both cards with Nighthawk in their name were cards from Zendikar. All in all a curious decision that doesn't really resonate with me. That said, if any guild represents cards coming back from the beyond, and this is a bit of foreshadowing, Orzhov is the place, so I won't hold your Phyrexian flavoring against you.
From a mechanical point of view, black and white are the two colors that deal in reanimation effects, which this sort of is. Black is easily the color most in tune with exiling as a cost, and white has various exile effects as well, though those typically aren't used on your own resources. That said, this feels like a very hard mechanic to get the numbers right on for a few reasons. First, this is a classic A + B mechanic. You
need exile effects, or this will do absolutely do nothing. This means you need to devote quite a bit of design space to support entry. Second, this mechanic runs the risk of creating repetitive board states. Phyrexian Nighthawk is a prime example, due to the self-exile clause. This card is a nightmare to get rid of, and I don't think it creates a desirable play pattern. Even other cards easily run into this problem if you have permanents that exile cards, like your rare Orzhov Arena. Third, these creatures almost by necessity need to be sub-par for the cost, simply because they keep coming back. How exciting is a whole guild full of
Persistent Specimens going to be? While that may be an extreme example, it's going to be hard to justify putting too much oomph into creatures with entry. Fourth, this mechanic is potentially really dangerous to print. Cards like
Food Chain do exist after all. I think you might have (consciously or subconsciously) realized this, since you put a life payment and higher mana cost on both your entry costs.
Going into the individual designs, I'm sorry to say that I've got additional gripes with all three of them. Starting with the uncommon, I think I've said almost everything I wanted to say about Phyrexian Nighthawk. The only thing left is that I don't feel like this is an uncommon. Since this self-exiles, you can always put it back onto the battlefield (provided you have mana).
Because entry has no timing restriction, this means you can plop down a deathtouch, lifelink, flying blocker at instant speed, making combat extremely frustrating for your opponent. (Edit: Future Onderzeeboot here. As Rusje rightly pointed out, you actually have to abide by the timing restriction of the card, since you cast it. Entry is not an activated ability. So, no flashing in a blocker at instant speed, meaning you can work around it for a turn with a removal spell. Still very frustrating to play against in my opinion, because it will come back on the next turn.) This design definitely does not feel like an uncommon, and I'm unsure it should exist even as a (mythic) rare without further tweaks. On the other side of the medallion we have Orzhov Arena. After reading the name, I was expecting a cool
Phyrexian Arena variant, but for a gold card, and a rare at that, it feels like the card does a whole lot of nothing. Yes it consistently fuels your entry game plan, but in a fairly lackluster and wordy way. In general, giving your opponent a choice makes cards worse than they read (see
Browbeat), and this is symmetrical too, so you end up losing life or paying extra mana as well. And that is after you gave away two Treasure tokens for free to your opponent. I'm sure this design gets some Johnny's gears turning, but for the average player it would be quite an unexciting rare to open. Your best design, by far, is your common. I'm not quite sure how often this offers a real choice, but accellerating you (at the cost of some life) or irritating your opponent is a cute choice. That said, with all of the life payment involved (both in the entry cost, and the upkeep trigger), this is certainly not a mono-white card. Unfortunately, it feels too wordy and too impactful for a common as well. I definitely would have bumped this up to uncommon.
I feel a bit bad devoting so many words to basically burning your designs down
It might not seem like it, but I hugely appreciate every submission to this contest a lot, including yours! Entry might not be a keyword I would personally pursue, but I think the Loan Officer has potential as an interesting custom!