Since we do not have the ability to see the specific language in your contract I am going to answer your questions in a broad form that applies to a general IDIQ contracting scenario. 

There is a common misunderstanding that Task Orders (TOs) and Delivery Orders (DOs) under an IDIQ are not contracts; however, TOs and DOs are in fact contracts.  The IDIQ provides the high-level requirement and the TO/DO issued under it, will contract products/services that support the higher-level IDIQ requirements.  TOs/DOs contain all the legal required elements of a contract and allocate government funds for specific products and services, qualifying them as an actual contract; which, support the higher-level IDIQ requirement.  

In order to utilize property across all DOs/TOs issued under a single IDIQ, you can easily include a blanket statement in each DO/TO providing the contractor permission to utilize any Government Property, on those DOs/TOs issued under the subject IDIQ.  A best practice that is becoming more common is the use of an enterprise TO/DO that contracts services and GFP to be used across the IDIQ platform.  Assets that are contractor acquired property (CAP) are acquired under the current TO/DO and then transferred to the enterprise contract at a later date. This can be done quarterly, semi-annually, etc., in accordance with the process laid out in PGI 245.103.   Items that are provided by the Government (GFP) can be immediately made accountable to the Enterprise DO/TO and used on the requiring DO/TO. Using one DO/TO under an IDIQ lessens some of the administration burden and provides a more efficient way of managing all assets under a single IDIQ DO/TO. 

To transfer accountability a Government asset to a different DO/TO, a MOD must be created for both the gaining and receiving contracts. REF for transfer of Government property: FAR 45.106, and DFARS PGI 245.103-71.