I think the wording is completely unambiguous with regard to the tumor reattaching itself to the alchemist - that is the tumor acting on its own, and therefore using its standard action.
However, with regard to tumor detachment, the text says:
"As a standard action, the alchemist can...."
It doesn't matter what comes nex in the sentence. If you diagram a sentence that begins like that, then it's the alchemist that is doing the thing as a standard action.
In this case, the standard action is "getting the tumor to detatch itself"
Once the alchemist uses his standard to get the tumor to detatch itself, it is not an action for the tumor to actually do the detaching, as no action type was defined for that part of the process.
Based on how the sentences are written - yeah.
The tumor can't act on its own, while attached, so it needs the alchemist to use its turn to empower it so that it can detach. Once detached it gets its own standard/move/swift actions each round. As such it is able to, without assistance, walk over to and climb the alchemist as a move action and reattach itself as a standard.