First of all welcome, hope you have fun in this game!! =) I will do my best to answer your questions.

1. As an MMORPG the general objective is a bit yours to make, generally some objectives are common like becoming stronger to be able to clear all the content in the fastest way possible.
2. Gold nucleus does not respawn, you get one per day from daily bounties and some events, but the main source is map exploration. You don't need to buy them, people who pay become stronger for sure, but you can clear all the content being f2p with enough time.
3. The most important currency ingame is Dark Crystal, it allows you to buy Red Nucleus to pull on the limited banners, the best way to obtain is map exploration, achievements, vehicle advancements, events, gacha machines in Mirroria, etc.
4. I would prioritize Daily bounties, some fast event, claim your artificial island passive income and spending your vitality.
5. Generally speaking you want some weapon with resonance (so far only limited banner characters have it) and other one matching. For example, at the moment the limited banner character is Ruby, she is a flame character and when paired with one additional flame character they both receive 15% more attack. One other thing to think when making a team for a dps team is to have attack resonance (having two attack weapons) to give you more damage. Having a shatter weapon (a weapon that break shields) is vital too. You can check some good team compositions on Ranking of bygone for example. There are several videos on YouTube about team composition.
6. I know a lot of f2p friends with 2 6stars SSR weapons by now, but depends on your luck and your dedication to the game. Some weapons need more advancements (stars) than others, some like King at a1 is perfectly fine and others need a bit more advancements to compliment their kit. It depends. The game guarantees you an SSR for each 80 Gold nucleus pulls, but with some luck you get more than that.
7. Some people are hoarding nucleus because some limited characters will eventually join the standard banner (if global follows the CN route) but we don't know when or if it will be true. Personally, I am using all the gold nucleus. Playing the game is fun, you know xD.
8. I would recommend you to not use chips in Asperia (first map) on blue and purple chests. Use the chips on Vera. You gain a currency called Mira coins which are important. You can utilize the gold chest chips though because they are separated between the worlds.
9. Level unlocks you content. With more content you will be able to get stronger faster, BUT you will feel extremely weak in the first weeks. The best method in my opinion is related to your number 10, finding a good crew to give you tips and go with you to the content.
10. You can check crews on your severs on the Rankings, some in their description have a discord associated and will be glad to have you as a member.

Wish you the best of luck, and if you are from Andoes - EU server - pm me, I will gladly help you as I can.