orbital mechanics presentation

[jevry]

So im holding a presentation about orbital mechanics for my exams and i need a tiny bit of outside help.

-In the presentations I want to explain about some basic orbital mechanics and rocketry maneuvers. The presentation should be around a 10 minute.
-For this presentation I also want to plan a fictional space flight to Europa that includes slingshot maneuvers (because that seemed like a great idea and my inner hipster did not want to go to mars because that seemed to mainstream).
I believe the correct route would be from earth -> venus -> earth -> jupiter?
-For this im assuming time is irrelevant and I can have any planet be at any point in it's orbit at any time (I will ofcourse disclaim this in the presentation). though I might want to know afterwards how long it may take for the planets to align how my route requires them, so then I can explain how the route isn't feasible (or is) because it will take too long (or won't) before we can launch the rocket.
-I don't need to explain all the calculations exactly (I don't think) but the more formulas the happier I make the examinators (or so im told), though at the same time it needs to be understandable and approachable. 

Turns out it's not as easy as ksp makes it seem
 

I've already figured out some of the basics of how to calculate and illustrate orbital trajectories. Things like the two focal points and one of those foci being the orbital body. i confuse the major/semi-major/minor axis (axi?) a little bit.
I'm not completely read up yet on excentricity and the time it takes to orbit a body
I do (I think) understand how you can calculate the distance with the velocity and vice versa with M_satalite*V_satalite*D_distance = E_tot 
One of my biggest sources so far has been this site: https://www.astronomynotes.com/history/s7.htm

In case its usefull here is a quick summary of how I so far think how I want to structure the presentation:

part 1: explanation planetary orbits
take Earth as example
Earth moevs around the sun
sometimes its a bit slower or faster and sometimes its a bit further away or close to the sun
an explanation how you can illustrate the orbit around the sun with some formulas and focal points

Stuk 2: doing orbital maneuvers with rockets
launch
reaching orbit and stabilising
escape velocity
orbit around the sun
changing course: slingshot maneuver with venus and earth
final course corrections
ariving at jupiter
arrivering at europa
circularisation
landing

i've also learned what N-body physics means and.... i mean it's interesting an all as this lad shows of here:

And I'm interested to see what will be up with the two planets in ksp2 that are "locked in a dance of death" since I heard those will get N-body physics (I may remember it wrong though).
But honestly it sounds like eldritch science not meant for the human brain to calculate and fully comprehend (and it probably is, indeed, only meant for computers to math it out)
and I don't think I need to calculate according to them in this case

 

so some questions i have so far:
-when performing a slingshot maneuver you gain speed equal to 2* V_p + V_r 
but is this different under different entree and escape angles ? does gravity, distance of the periapsis, and speed of the planet influence this? how can i calculate this and why has this not shown up in explanations before?

-during interplanetary transfers, should i take into account that the planet you are escaping from will nudge your orbit a bit? (your periapsis (assuming your destination is farther away from the sun than earth) will be a bit higher than that of earth if im correct)

-does anyone have any advice where to find calculations I need for the route im planning?