If we kept ourselves divided during space exploration, would there be wars over the control of Mars and the Moon ?

No, there is too much of a distance between the two plus I don't think anyone could be bothered
 Anyway the AIU made up a Space Treaty to cover such a thing
 The Space race was being led by the Russians at first
The USA needed to come up with something special
 The Moonshot won hands down
A lot has happened since, the Cold War is over, and many countries work together now
Using Russian Rockets by the Way
As a Footnote, many of the Stacks that were to power American Nuclear missiles were sold to the Russians and they use them in their rockets
Russia was always on the Back foot where Rocket Technology was concerned, so USA rockets were welcome
In 1968 Russia built the N1 Rocket to take Cosmonauts to the Moon before the USA, but through outdated Technology, a rush Job and filling it with Oxygen only
It Exploded on the Launchpad while building
It was Russia's loudest Peacetime Explosion

Nah.  We signed a treaty to the effect that no one can own any of the heavenly bodies - kind of like how we regard Antarctica. 

There will still be wars over the rights to the moon and or, whatever else we achieve in space.

No doubt.

The military is an institution with an insatiable appetite for funding. The bigger it is, and the more it can instil anxiety into the population via the media, the more funding it can demand to fuel the expansion it claims is necessary to match "the adversary", either real or fictional.

The United Nations 1967 Outer Space Treaty was modeled on the Antarctica treaties. Yes, there have been incidents in Antarctica, including one attempted murder by stabbing. I think one incident did end up going to the international court at the Hague in Brussels

Very likely both the Moon and Mars are going to be like Antarctica has been for decades, also for decades.