Re: do the cytosolic proteins touch DNA during mitosis?while nucleous is broken

Actually, nuclear pores do allow and facilitate the transport of proteins 
into the nucleus.  Proteins containing a nuclear localization signal 
within their amino acid sequence are efficiently translocated from the 
cytoplasm into the nucleus.  For example, signalling hormones bind to and 
activate hormone receptors in the cytoplasm.  When the receptor is 
activated, a chaperone protein is released from the receptor to expose the 
nuclear localization signal.  The hormone receptor is then translocated 
into the nucleus where it binds to DNA and stimulates transcription of 
specific genes.
To answer your question as to whether cytosolic proteins touch DNA during 
mitosis, the answer is yes.  To give just one example, the centrosome 
which resides outside the nucleus is a complex of many proteins.  This 
structure nucleates microtubuleswhich extend and contact the chromosomes 
once when the nuclear envelope breaks down.  These microtubules will then 
shorten and pull the chromosomes apart.

Alberts, et al,  Molecular Biology of the Cell, 3rd Ed. Garland Publishing 
NY, 1994, pages 566-567 and 911-933.