Items you pick up in one round and store for next round, will count as your items in next round and you will pay upkeep for those items.
About avoiding upkeep.. spawn, kill some enemies, wait until there's 2 players left, leave the server, come back for next round and you won't pay any upkeep. You won't be getting multi either but if your team is about to lose it's better to leave to avoid (most certain) high upkeep than stay for a couple ticks. Just play next round when your team is winning and you'll be able to upkeep pretty much anything without losing any gold.