If I have Switcheroo at Trifarrian War Camp, and I switch Irelia with another unit, how does it work? How does it work with the buff from Battlefield and also the Choose Me?

Here's how it all works.

Yes, playing Switcheroo on Irelia triggers her +1 Might. Switcheroo chooses/targets two units (Rule 433), and Irelia, Fervent's ability says "When you choose or ready me, give me +1 Might this turn." So she gets +1 from being chosen.

You control the order on the chain. Since you control both Switcheroo and Irelia's trigger, you decide which goes on the chain first. The last item you place resolves first (LIFO). If you put Irelia's trigger on top, it resolves first (she gains +1 Might), then Switcheroo resolves using the updated Might values.

How the numbers work with Trifarian War Camp: The battlefield gives +1 Might as a continuous effect on all units there. When Switcheroo resolves, it calculates the difference between the two units' current Might (which already includes the battlefield's +1 and Irelia's +1 from being chosen if that resolved). Then it applies a -X modifier to the higher-Might unit and a +X modifier to the lower one for the turn. All continuous effects (battlefield, Irelia's trigger, Switcheroo's modifier) keep applying on top of each other through layers.

Example: Irelia (4 base) and a 3-Might unit, both at War Camp (+1 each). You choose the order so Irelia's trigger resolves first (+1). At Switcheroo resolution: Irelia = 6, other = 4. Difference = 2. Irelia gets -2, other gets +2. Final: Irelia = 4 + 1 (camp) + 1 (chosen) − 2 (swap) = 4 Might. Other = 3 + 1 (camp) + 2 (swap) = 6 Might. The continuous effects mean neither unit gets "locked" to a number — if either later leaves the camp or gains more Might, the modifier adjusts on top (FAQ #9775, FAQ #8531).