Yes he has access to 20 m SP, no he cannot fly a perfect slicer. TLDR the 5 million SP limit is "free training limit" and the 20 million SP is a hard "access limit" to a specific set of alpha skills, meaning some skills for navigation, tank, SC command, etc, will be below 5 even if trained above that.
HOWEVER! At current prices, what I recommend for most returning high SP players who want to come back to the game, is dip your toes in, if you really want, I am sure you have at least half a bill in ships or ISK around. Get a skill extractor or two, extract up to two million SP (while alpha status it will have to be those Omega accesible skills only) then sell the injectors for PLEX and a couple new extractors, that PLEX will earn you back somewhere in the neighborhood of ....2 Mil SP!
At the current price, if you stay in HG implants and buy/sell direct to the market, you will come out about even, if you set up a buy/sell order bc you have the patience and ISK to do so, you can earn a tiny bit of extra ISK. basically you trade the ability to earn any new SP for access to all the SP that is blocked by your Alpha status.