Of course you can do it yourself, you have the right to provide the VO on your own initiative with any information that facilitates the processing of your case, including medical

You can indeed gain a month or two by doing it proactively. Usually, it's requested right after the interview or just before they want to make a favourable decision without an interview.
Are you in Canada or not? In any case, it basically comes down to a timing game. A permanent resident visa, when issued, is valid for 12 months since the day of the X-Ray in the latest medical exam report they have on file.
So, it's your call. As a general advice:
If you're outside Canada: 1) If you are pretty sure the decision will be made within a reasonable amount of time with enough time of visa validity remaining for you to enter Canada, then go ahead.
2) If you think there's still many more long months to process your case, then you might want to hold it off for some time...
If you're inside Canada, the above stands true, but isn't so relevant, as long as there is almost ANY time left of the 1 year after the medical when you get your passport with visa back.
In the worst case (outrageously long case processing) you'll have to re-do your medical...
I can't guess what the processing time's going to be for you guys. CIC website says, 80% of cases are finalized in Paris within up to 8 months...
P.S. In the case of my wife and son, they did the medical soon after submitting our application package but the X-Ray part of the report was done a few months later, because she was pregnant during the initial examination... It was an inland application.
Hi Kazakhstan,
Appreciate your answer on this matter

We are both still in Europe, planning to move soon though.
I have no idea if I will be asked to attend an interview, hope not...but we would like to gain some time and we believe the PR will be granted.
As far as how long it takes for the Paris VO to process the application, no idea , I know they indicate 8 months, but they deal with many different countries and maybe it also depends on my nationality. We have a very straightforward story, we met at work, we work together, actually, since we filled the application, we moved countries, due to our work, and again we live together.
But the all process sounds scary to someone that is just not used to all of this...