The tutorials around IOTA are the best way for developers to have a deep dive into how IOTA works and what it makes possible.
Have a question or you're not sure where to start? The FAQ is the right way to begin your new journey in IOTA.
Rebroadcasting is basically sending the exact same transaction to all of your neighbors again. Most of the time, apart from a situation where you're sure that your neighbors didn't get your transactions, replay is more useful to get your transaction accepted by the network.
Replay is the process by which you "re-attach" your transactions to a different part of the Tangle, in the hopes of getting your transaction accepted. As such, you are completely redoing the transaction process (tip selection and Proof of Work).
At the time of writing, there is no way of running a private Tangle - yet. The IOTA Foundation is currently establishing a framework with a set of tools that will enable you to run your own private Tangles. Currently you can use the Mainnet and Testnet, which should be well suited for any of your testing purposes.
When interacting with any of the libraries which are supported by the IOTA Foundation, you can be made sure that your seed will never leave your side. All the essentials, such as signature generation, validation and so forth is done client-side, meaning that it's done locally on your environment. As such, your seed is never sent to your remote node and you don't have to worry about any security issues.
The IOTA GUI makes it possible to choose between Full Wallet, and Light Wallet. The Full Wallet automatically runs an IRI (IOTA Reference Implementation) instance in the background, which in turn means that you need neighbors in order to participate in the network and be synchronized. If you run the full wallet, you have no "trust requirements", as you are completely and independently participating in the network.
The Light Wallet makes it possible to connect to a remote node (whether it is your own node, or someone else's), and it's uses to get the latest state of the network, most importantly branch/trunk transactions which are required for making a transaction. When running a Light Wallet, it should be noted that your seed never leaves your wallet, and all the sensitive work (such as signing) is done client-side.