Once you have an item number, you can use it to search eBay's website for that specific item. It can be done anywhere the main search bar is visible, or by going to 'Advanced search' and choosing the option 'Sort by item number'.

The transaction ID is something that is assigned by eBay and should be easily recognizable in the transaction details they are furnished by eBay at checkout. PayPal also assigns a transaction number if they paid using PayPal. This information would also be available to them through their PayPal account.

If you purchased your item as a guest, the order won't appear in your purchase history. You can find your order details in your order confirmation email.

If you mean the eBay item number, yes you can. Click the advanced word just right of the search button. You will then have to search sold/completed items to find it.