Hello CuriosityX - and welcome to the forum
I fully understand and appreciate your (good and pertinent) questions, so let me try and address them to the best of my explanatory capacity. 
Re: your 1st question: did Tycho convert his numerical records of the planets into a small scale orbits (as seen in the image that shows Mars trace over 16 years)? or was this done after his departure? .
The image you mention is the diagram attributed to Kepler titled “De Motibus Stellae Martis” (“Of the Motion of the Star Mars”) - and was published in the first chapter of his “master treatise” Astronomia Nova (1609). Here’s what we can read about it at the University of Chicago Press website :
“In the first chapter of his Astronomia Nova , Kepler introduced a diagram that, he claimed, represents the path of Mars between 1580 and 1596, assuming a resting Earth. In the words of William H. Donahue, “the appearance of this diagram is a dramatic moment in the history of thought. Nothing like it had ever been published before.” It is unique mainly because it is the first diagram that offers what is explicitly asserted to be an accurate representation of the trajectory of a planet, the first diagram in which the path of a planet appeared clearly separated from the geometrical devices that produced it. This essay aims to offer an analysis of the accuracy of this diagram. The authors try to determine what tables, ephemerides, observations, or models are behind its construction and how accurately the diagram reflects the different elements of the trajectory of Mars. The outcome of this analysis raises the question of whether the diagram was made for the role it ultimately played in the Astronomia Nova or for some other purpose.”
Here’s an image from Chapter 5 of my book which shows just how precisely the Tychosium 3D simulator matches Kepler’s (or Brahe’s?) ‘geocentric’ diagram of the motions of Mars between 1580 and 1596 :
Astronomia Nova was published about 8 years after Tycho Brahe’s death - and we know that Kepler stole all of Brahe’s papers and drawings, so we simply don’t know whether Brahe was aware of
this diagram - or if he (possibly) even was the original author of it!
Re: your 2nd question: Do we know what was the exact task that was given by Tycho to Kepler?Like was there something baffling Tycho specifically about Mars to ask Kepler’s help?
Well, Brahe was obviously aware of what we call today the General Precession, i.e. the constant eastward drift of the entire star firmament, currently reckoned to amount to about 50.3" arcseconds per year. (Incidentally, and quite interestingly, Brahe’s estimate of the GP was 51" arcseconds, which compares well with my own estimate of 51.136" arcseconds in the TYCHOS model).
But here’s the thing: since Brahe believed that the Earth was immobile, he would have either attributed the observed general Precession to:
- A general precession of all the stars - all of them drifting eastwards in unison by 51"
or
- The infamous Copernican idea of a slow “wobble” of the Earth’s axis
Being a very smart fellow, he probably didn’t like the 1st option (a general drift of all the stars). This left him with the Copernican idea of the Earth’s “lunisolar wobble”. Yet, since this wobble would also have affected Mars, making it drift eastwards by 51" every year (which it does NOT), he would have been deeply baffled about this - since his wondrous observational instruments (reputedly capable of a 2" precision) certainly would have allowed him to notice this discrepancy - over the many decades of observations that he made in Hven.
In short, Brahe was probably ‘stuck’ with this problematic affair - and thus allowed Kepler to ‘give it a go’ with his mathematical skills (although the two never liked each other and kept arguing & bickering - as widely reported in history books).
All in all, it all comes down to the fact that Brahe was not aware of the Earth’s translational motion around its PVP orbit amounting to an annual ‘westward’ / annual displacement of about 14036km. Now, Mars returns “in opposition” - as it passes closest to Earth - every 15 or 17 years, during which the Earth will move ‘westwards’ by a good 210450km or 238612km. Tycho Brahe’s fine instruments would surely have ‘detected’ these earthly displacements (which would have upset any calculations and predictions that Brahe would have made of Mars’s motions), yet the poor man would not have known what to make of it all !
