How good is your English?
Do you ever plan on visiting Europe?
Why were the Lunarians so afraid of human astronauts when they could easily defeat an army of youkai?
Why is Byakuren's hair gradient?
-Nonexistent
-He went on a Europe trip to Germany -> Czech -> Italy just last year-I don't think they're literally afraid of them, but rather just don't want the moon to be tainted with impurity. The Lunarians don't want to fight but apparently would if necessary, implying that humans merely coming to the moon poses a problem, which is pretty much what everything else points to anyways.
-Originally he wanted her black-white clothing design but it looked too depressing so he gave her blonde hair. Then he thought it was too bland so he gave her the gradient.
What is the basis of the season numbering in Gensokyo? In particular, what is the relation between the season numbering and the creation of the Great Hakurei Barrier? In addition, when does the season number change?
Was Remilia in Gensokyo during the Vampire Incident?
Might as well ask. After all, for some time it was claimed that Remilia entered Gensokyo after said incident, citing Akyuu.
It's a lunisolar calendar derived from the japanese calendar which was really derived from the chinese calendar. Gensokyo uses the creation of the Hakurei Barrier to begin the year numbering (Year 0), and they instead call the years "Seasons" (the term 季 can mean year). Of course, they still have the four seasons per... season.
They use a sexagenary cycle, which was also derived from the chinese calendar. However, instead of the chinese's cycle of ten Heavenly Stems and twelve Earthly Branches, it uses a cycle of the "three lights" (sun -> moon -> star), a cycle of the four seasons (spring -> summer -> autumn -> winter), and a cycle of the five elements (earth -> fire -> water -> wood -> metal). Each of these increment every year. By the Chinese Remainder Theorem both of these systems have unique combinations for every year, for sixty years:
Chinese: 10 = 2
1?5
1, 12 = 2
2?3
1 -> lcm(10,12) = 2
2?3
1?5
1 = 60 -> years have unique remainders mod 60; similarly
Gensokyo: 3 = 3
1, 4 = 2
2, 5 = 5
1 -> lcm(3,4,5) = 3
1?2
2?5
1 = 60 -> years have unique remainders mod 60.
This cycle is described first in A Beautiful Flower Blooming Violet Every Sixty Years. You can get the current year in the cycle by taking our year in CE, minus 25, mod 60. Like this year is (2013-25)%60 = 8, which is the Year of Star, Spring and Wood. To get each individual attribute you can take the year minus 25, mod 3, 4 and 5 respectively.
Remilia was "the vampire" in the Vampire Incident. I'm not sure where you think Akyuu says otherwise, but whenever she refers to "the vampire", she does mean Remilia. It's made clear when she says the first time the spell card rules were formally enacted was during the Scarlet Mist Incident with that vampire, and is further supported during other works when Remilia and Yukari refer to contracts.