Flash is the most power full summoner sometimes contested by ghost for some champs. I like flash on all in my team.
Cho need teleport to be able to join teamfights and maybe sometimes counter a bot splitpusher. The splitpusher of the team will usualy be Poppy though.
Poppy need his smite for jungeling.
Ryze need to survive dives early and mid game. Heal or barrier helps alot.
Trist need heal or barrier too as ADC.
With Trist's explosive shot active I don't think ignite is important on support. I preffere exhaust on my support.

Tankiness is balanced with some damage output on all champs. Trist is the only real squishy target, and she will be very hard to get to with her range, E, and R. She is up there with Ezreal when we talk safe ADCs. Ryze will probably be near her to protect her with W (Rune Prison) and the rest of the team all has excelent tools to zone, stop or kill divers.

Flair is very nice sustain and magic damage scaling infinitly. Very nice dps item late game. Early it's a lot of gold not doing much though, but it will provide more gold than the other jungle items since the free ward almost has the same value as there gold genneration.

Gaunlet is the mandatory sheen item I choose. It's the armor item in this build and it makes it very hard to kite Poppy both with the slows and the CDR. Once you complete it clear speed and split push will be fine.

CDR boots are also anti kitting and does wonders for your dps. Combined with masteries, blue pot and Gauntlet this will cap the CDR.

Maw is amazing with poppys passive. A 400 shield at low health probably has the same value as 600ish health. You will survive for a surprisingly long time at low health making the AD on the item much more valueable to poppy than other champs.

Blade would not be my first choice if it was not for Flair since it does not provide much AD for your Q damage and Q is an attack reset. Poppy have good use of a second attack speed item for boosting the auto attack damage output from Flair. The active is also super nice excecute, additional antikitting and sustain. With my masteries the heal at low health can do wonders - again Poppys OP passive synnergize nicely.

Void staff provides the penetration that you will need late game with no pennetration from runes or masteries. Early you enjoy the bennefits from beeing mixed damage making it hard to itemize against you, but late game magic penetration will boost both Poppy's Qs and the damage from flair. You might find that this is not needed and you have plenty mixed damage without penetration, but I like it better than additional tankiness.

Trinity Force alternative?
Many people preffere Trinity Force on Poppy as her sheen item and true - it will provide many desirable things. The mobility is nice, but synnergize badly with poppys W due to deminishing returns (if you want you can buy alacrity). The attackspeed will synnegize nicely with flair and the combined stats is alot more damage. The problem is that I see no alternative to having to swap Blade or Void staff with Randuin's to round off the tankiness of the build and then you no longer deal more damage, you are much easier to kite, you will clear the jungle slower, splitpush slower and you no longer have capped CDR. The active on Blade will have to make up for the lack in HP and works better with Poppy's passive than HP if used well.