When I start my full node, there are multiple offline nodes that bitmonerod keeps attempting to connect to.
How can I manually stop bitmonerod from trying to connect to nodes that are no longer working? If I do nothing, exactly how long does it take before offline nodes are dropped by the network?
If I recall correctly the IPs of these nodes are stored in p2pstate.bin and therefore bitmonerod (the daemon) will try to keep connecting to them. This also answers your second question. There are two solutions two solve this issue. First, you can delete p2pstate.bin. Bear in mind to exit bitmonerod first before doing this. Second, you can manually ban these offline nodes with the ban command in bitmonerod. This command allows you to ban a certain IP for a certain amount of time.
Automatic IP baning is still not present in Monero. Boolberry (and later AEON) adopted an ip banning mechanism for misbehaving nodes. Multiple offenses are required to get banned but some DoS type of attacks result in immediate ban.