Indexing manager missing, search fields like multilist is not displaying windows

Hi Olga,
I believe I have found the cause of your issue. It seems you are missing the sitecore.buckets.config file. It looks like it was either disabled or it's properties patched out. The reason the issue occurs is that ContentSearch is highly dependant on item buckets. This means thats when Item buckets are disabled, some core functionality with content search will be broken, therefore it is also disabled to prevent any confusion. This was introduced in the 7.2 Update-1 release which you can see here 
https://doc.sitecore.com/sdnarchive/SDN5/Products/Sitecore%20V5/Sitecore%20CMS%207/ReleaseNotes/ChangeLog/Release%20History%20SC72.html#72update1
Here is the specific note in the release:
New features

• It is now possible to disable all item buckets and Sitecore.ContentSearch functionality using the new BucketConfiguration.ItemBucketsEnabled setting in the Sitecore.Buckets.config file. (398925)
• When this functionality is disabled, Search tabs do not appear in the Content Editor or in dialogs, all item bucket commands are hidden in the UI, the Indexing Manager is not available in the Control Panel application, and the BucketManager API ignores calls. In addition, all background processes related to indexing are disabled. (398925, 403858, 400821, 399186, 403852 )

Therefore to fix the issue, please reenable item buckets and the indexing section should reappear.
Best regards,
Phil.