How to capture a screenshot without the own character in the picture

I did some testing by myself in the Testing Range on Xbox 360, killing myself via grenades or various hilarious helicopter mishaps and then seeing what persisted after respawn. Here's what I learned:

Despawns on death: First Aid Pack, Medic Bag, Ammo Pack, Ammo Box, C4.
These disappear when you die, no matter what loadout you choose next. These may persist until you respawn, I couldn't test that by myself. The C4 is a bit weird; it occasionally lasts a second or two after respawn and if you're fast enough you can even detonate your last spawn's C4. My guess is a timer starts to remove it when you die.

Persists as long as equipped: EOD Bot, MP-APS, Radio Beacon, T_UGS, SOFLAM, MAV, Mortar, AA Mine.
These stay in the world where you placed them after you die, and you can respawn and re-connect to them. If you remove them from your loadout or change kits they disappear.

Persists forever: M15 AT Mine, M2 SLAM, Claymore.
These stay in the world where you placed them even if you spawn with a different kit. Note that there are limits to these items, like only 3 mines of each type. If you drop a 4th mine the 1st despawns.

bobfet1 already said it:

It doesn't look like there is a good way to self-cast an ammo pack.

You can adjust the range your soldier throws the package. If you look closer to the ground, you throw it in front of your own legs. This is how you can supply yourself without moving. Keep in mind that you don't have to stand exactly on your supply, it has a small range. Others, by the way, can also get supplied by this, so if you want to supply only yourself, keep in mind you need a little distance to your mates.