With respect to this particular discussion, I'm trying to get a sense of whether or not my request is legitimate. And if it is not, why isn't it legitimate?
Asking for a sign was never respected by the Lord.
"But He answered and said to them, An evil and adulterous generation seeks after a sign, and no sign will be given to it except the sign of the prophet Jonah." Matthew 12:39
God will reveal Himself to you through other ways if you are open to Him.
"Since what may be known about God is plain to them, because God has made it plain to them. For since the creation of the world His invisible attributes are clearly seen, being understood by the things that are made, even His eternal power and Godhead, so that they are without excuse" Romans 1:19-20
If you are humble enough to listen to other believers who once were where you are now, the Lord may also reveal Himself to you through them.
"We are therefore Christs ambassadors, as though God were making his appeal through us. We implore you on Christs behalf: Be reconciled to God." 2Corinthians 5:20
God is pleased with faith.
"But without faith it is impossible to please Him, for he who comes to God must believe that He is, and that He is a rewarder of those who diligently seek Him." Hebrews 11:6
If you really want to know the Lord, you have to have your heart in it, not just sit back and "think of a number." This may involve humbling yourself.
"You will seek me and find me when you seek me with all your heart." Jeremiah 29:13
Salvation is a gift which you can receive by believing.
"For by grace you have been saved through faith, and that not of yourselves; it is the gift of God" Ephesians 2:8
After receiving salvation, the "sign" which you sought in the beginning now lives in your heart.
"in whom also, having believed, you were sealed with the Holy Spirit of promise, who is the guarantee of our inheritance until the redemption." Ephesians 1:13-14
The Kingdom of God has been called the upside-down Kingdom. Why? because everything is the opposite of how we think it should be. To save your life, you have to lose it. To be strong, you must be weak. To be exalted, you have to humble yourself. The last shall be first and the first shall be last.
The Jews expected God to come in power and reign on earth, but He was born very lowly in a manger, lived as a servant, and died a humble death. You expect God to give you a sign in order for you to believe, but God will give you plenty of signs after you surrender to Him, including his Holy Spirit.