No, you only need to meet the goals on an exploration card to acquire the card and move it to your Points Area. The rules clearly indicate that you don't have to spend experience to complete missions. You should treat experience as a pool though. When you spend experience to gain cards from the Starbase or to repair damage, that experience isn't available for purchasing any other cards or repairing other damage. The glossary doesn't define what it means by spend, but this is clearly the intent.
This is covered in the Rulebook.
Q.The Mission I want to complete, has a goal of 6+ XP, do I have to spend 6 XP to
A. No, you just need to have cards in your Bridge that combine to 6 or more XP. That XP
can be spend for other things as usual.
Yes, you only need 4+ Attack to complete the goals of those missions. Normally, you only get one Explore per turn though.
GOAL - This is the criteria that must be met to acquire this card and move it to your Points Area
Perform 1 Explore - The Space Deck contains several different cards that you must
obtain to gain Mission Points and win the game. To explore, you flip over the top card of the Space Deck and proceed based on the type of card flipped (See page 15 for Exploring the Space
Deck). This can be performed multiple times if you get