I'll just jump right in here:

1) Would it be possible to remove the animations for 'obtaining' and 'completing' a magian trial, ie getting hit in the head with stars and getting hit with hearts? While these animations were exceedingly cute the first, I don't know 5 times, they're just cause for more wasted button spamming after about 100 trials.
Possible solutions: Make it so only the first and last trials result in these animations.
Make it so you can 'toggle' the animations on and off, like skipping a mission cutscene or some such. Like making it an option if you just talk to the magian moogle he asks if you wish to hear his instructions on every trial.

2) Running back to the magian moogle himself when doing weapon trials.
I know this is a relatively minor annoyance, this completing a trial, stopping everything you're doing, leaving the zone your next mob may be in, running back to ru'lude and running ALL the way back to the zone you were JUST in, and having your weather wear off while you were doing all this just to get annoyed by stars and hearts, but i think we could stream line it just a tad.
Possible Solutions: Make it so when you go to choose your trial you are actually choosing the 'path' of the trial, not a single trial. For instance, you go the magian box and choose an axe, trade to moogle and he asks what trial you want to undertake, you choose the one that ends with farsha, instead of coming back to him between every NM etc, it commits you to that end, automatically updating your magian log to the next NM set in that path, or item acquisition, so on and so forth. For trials which diverse at certain points and you're not sure which path you want to take, it could be possible to pick at which point along the trial you want to stop and re-evaluate which way you want to go. The moogle already knows all the possible divergences and what trial ends at what.

I don't know if these 2 things would even be possible, but they would save a lot of people time and warp scrolls.