You should get, at the very least, something that has lvl 10 weps. From this point on, it gets progressively harder.
Level 6 weapons will not cut it, as they won't even kill the shields. There are plenty of ships available that have 2 level 9 weapon slots and 4 level 10 weapon slots. There are even some ships that have 6 level 10 slots.
Here is the page to look for a ship that will suit your needs. https://swat-portal.com/forum/…/377-Very-Heavy-Fighters/
Good luck in your search.
~edited to answer second question~
Yes, there are breaks in the chapters to allow you to go on a search for a new ship, equipment, exploring, etc. However, there are only a few of these breaks.