Find matching fields between content types and create new nodes
You need the below rule to fire when the client's content type is saved.. And all relevant fields must exist in BOTH content types. It will talk your clients answers, feed them through a VBO, get the company NIDs that match and then loop through them to create Partner Details nodes.


First, you need to have Views Bulk Operations (VBO) which will give a new rule action.. "Load a list of entity IDs from a VBO View". In this action, you can specify a VBO view and a list of arguments (taxonomy tids etc.) to send to the view using replacement patterns. The list of returned IDs is what you will use to create your "partner details". 
You need to create the VBO view by adding "Bulk operations: Content (Content)" as a field in a new view. The arguments sent from the rules action can be picked up by contextual filters using the "Provide Default: Raw Value from URL" setting. If I remember correctly, the first argument starts with path component 2.
The results of the VBO should be limited to those nodes that match the arguments. Now we have a list in rules that we can use Loop to go over. 
Depending on needs, you can just create the new entities and reference the client and company nodes or do a two way reference so that the client and company nodes also reference each partner detail node.


The above is will do the taxonomy fine but for the integers, it's a bit more complicated since the contextual filter must match perfectly. For this, you can use the contextual filter: Global Null (which won't limit the results) and use Views Argument Substitutions to take the value from the contextual filter and use it in your views filters (the regular ones on the left). * *  *!2 * * * (no spaces) if your global: null is the second contextual filter.

This is pretty limited but it could work. If you need it to only be 50-100 employees for example,  on the company content type, have minimum and maximum integer fields and use "greater than" and "less than" filters in VBO view.

You should do quite a lot of testing on the VBO preview to make sure things work as expected. I really believe this is the only way to do this without writing a nasty amount of custom code. Good luck!