Yes. These can change during transmission.
Options field changes. For example when you use the record route option, with every hop the options field records a new router address, so the options field of the packet will change. Here header length will also change because overall header size is increasing with each newly added address.
Now for fragment offset.
Suppose there is a fragment A with fragment offset X and length L. It will remain X until it is further fragmented. When fragmentation happens, new packets (fragments) are created and the same IP header is appended to each fragment with few changes. Every fragment will have a different offset.
So for a fragment A with offset X and length L, offset cannot change but further fragmentation can create new fragments out of this fragment A and those fragments will have different offsets not all same as X.
