Techniques to show a protein heterodimer-DNA interaction
How will you check that fra1 is expressed or not in these cells.


RT-PCR is a quick and an easy method to check for expression. However, it only tells you about the RNA level expression and you would have to work with the assumption that RNA concentration is directly proportional to protein concentration.

Considering that you do not have an antibody for fra1, you can go for mass-spectrometric protein quantification. It is good for a high throughput analysis and is generally wasteful when you just want to study one protein. However, people do carry out targeted proteomics to study a few proteins. 

Alternatively, with some effort you can actually develop an antibody for fra1.


  How will you check that the heterodimer bound to the given DNA
  contains fra-1 along with c-jun.


Showing ternary complexes (fra-jun-DNA) are a bit difficult. It is relatively simple to show pairs. You can show that two proteins form a complex in vivo by doing a co-immunoprecipitation followed by detection by either western blot or mass-spectrometry. 

You can also show interaction by biophysical techniques involving FRET and SPR but these cannot be performed in vivo (at least SPR. FRET can be done but there are limitations).

You can show the DNA-protein interaction by ChIP-seq as you mentioned in your question.

Showing all the three together is a bit tricky.

You can do it using in vitro assays that employ fluorescently labelled (one GFP and other RFP; or FRET pairs)/custom epitope tagged (for e.g. one FLAG and the other HA -tagged) proteins and a known DNA sequence generated by PCR/DNA synthesis. You can use EMSA along with antibody/fluorescence based detection to show the ternary complex.

You can indirectly infer the presence of the ternary complex when the ChIP-seq using antibodies for both the proteins shows up the same DNA sequence in the reads. You can also express a FLAG-tagged (or any other tag) protein, if in case you do not have an antibody for the proteins.