If a wallet is down except for a few hours each week to catch up the block chain, is the wallet's IP address associated with that wallet's transactions? Can others associate a wallet's receiving address with its IP address?
You are making a common misunderstanding of how Bitcoin works.
The Bitcoin client collect IP addresses and the Bitcoin blockchain doesn't store them. From the data in the blockchain there is no way to associate a transaction to an IP address or to know which node is responsible for mining each block even.
This misunderstanding comes from the information collected and reported on the third-party service, Blockchain.info. They have a backend that makes many connections (over 3K at present). When a node that blockchain.info's backend is connected to is the first they know of to see a transaction then their service tags that transaction with that node's IP address. But there are tens of thousands (or more) peer nodes on the Bitcoin network so very often this "first relayed by" node IP address information is inaccurate. If your node does not accommodate incoming connections, then quite often the transactions your node makes will be reported on Blockchain.info as having been "first relayed by" with some other IP address than your own.
This same IP monitoring is part of the approach that Blockchain.info uses to try to determine which node (mining pool) is responsible for solving a block. And that's why there is a sticky at the top of the mining forum board. An excerpt from that sticky:
If they are connected to an especially fast and well connected relay they will falsely report it as being the 'source' of many blocks. This doesn't mean that this relay is a big miner, or that they're attacking the network, or anything like that. An example of this is the Swiss node at 82.130.102.160 but there have been a number of similar misattribution addresses in the past.
People freaking out over these misconceptions will now be shot.