Drifting weapon spawns:
In Halo 1 Xbox, the game ran in lockstep. So each Xbox ran it's own timer and spawned weapons back on it's own. So they never drift since the weapon spawns will never be more than 1 frame apart on each box. Each Xbox has the same processor and clock so they can depend on that.
In Halo 1 PC, the host has to sync the power up spawns. So it requests each client to spawn a powerup in the spot. This takes a bit of time. Then the client that picks up the powerup has to tell the server they picked up the powerup, which will delay the timer being restarted a bit. Clients in a session could either be in 30fps or 60fps and all have different quality clocks (in addition to the host), and you have to do time correction (much more server when catering to dialup).
Delayed respawning:
In Halo 1, clients request a spawn location and tell the server/other boxes where they are going to spawn. On Xbox this was instant due to being in lockstep, so you don't notice it.
In Halo PC, you have to ask the server, the server checks to see the space is open, then tells the client it's okay to spawn, then the client spawns. Due to latency, it takes a bit to spawn in.
In Halo 2, 3, Reach and 4, the HOST controls your spawn, and tells you where you're going to spawn, so respawning is a lot faster, and also why you can be shot at before you're fully spawned in on your box because the animation hasn't finished. The host basically knows where it's going to respawn you before you even die.
The exception to this is Invasion in Reach. Since you could choose your spawn, the host had to wait to see what spawn you wanted, check to see if there was enough space and safe area to do so, then spawned you. This is why in Invasion it was always better to stay on your original spawn or choose your spawn as soon as possible and NOT toggle between spawns, because each time you toggle the safety checks were re-done, and that's why you'd be stuck at "Respawning in 0.." for a moment