mongoose traveller learning new skills: I guess I probably haven't really discussed this before. Basically, learning new skills in MgT is slow, but doable. Raising a skill from level n to level n+1 takes a number of weeks of training equal to the total of all of your skills plus n+1 (to gain a new skill at 0, treat n as -1). Since about the only training you get happens in jump, an experienced character is like to pick up maybe 2 new skills over the course of a short campaign like the ones we play. On the other hand, if you start the campaign with a rank newbie with a skill total of 3, you learn really damn fast, and since you get to choose your training skills (but not your skills from character generation), this also provides a character who can do exactly what you want him to be doing. Avoids the risk of aging, too, but you lose out hard on material benefits.
As an aside, we have a houserule that if you roll snake eyes on a check with an untrained skill, it counts as a week of training. Experiential learning, if you will.
mongoose core weight: I think this is referring to encumbrance, probably (since I don't know what the actual paper copy of the Core weighs). The rules for it are on page 104, and I highly recommend using them. Otherwise everyone ends up in Combat Armor with laser rifles and things go all to hell.
mongoose traveller where does the ship exit jump space?: So, there aren't actually hard and fast rules on this. The only guideline I know of is from page 141:
When the ship exits Jump space after an accurate Jump, it tendsThat's all there is on it. Jump accuracy is determined on the bottom of the left-hand column on the same page.
to arrive close to the target world, but outside or on the verge of
the hundred-diameter limit. Inaccurate Jumps just dump the ship
somewhere in the inner system, requiring a long space flight.
Short version: Wherever you want them to, with the guideline that success means close and failure means far.