There is nothing more ignorant in my opinion than no one responding to a poor chap's plea for assistance…
So, as a long-time player of AH, I'll give you a bit of feedback on your Investigators:
Ash: For both Boomstick and Shop S-Mart, there's far too much accounting going-on too make this worthwhile. For Boomstick, it may be better to provide a flat +2 Combat bonus when using the Shotgun, than having to go through the machinations of calculating whether its higher or lower than a specific amount. For Shop S-Mart, I do not see the connection between the Investigator and the ability.
Doctor: The Tardis ability appears fine, as it's something useful, but not game-breaking. As for Regeneration, I'm not convinced that this one is worth the trouble. Also, a "7" in any stat is way too high for an Investigator.
Al: Recommend changing Difficult to Catch to simply a roll and if it's a Success (Normal 5-6; Blessed: 4-6; Cursed: 6) instead of having it as a 50% every time. Also, Stray Shots is too bulky…recommend saying he's able to ignore the first attack if he fails an Evade check, for example.
Francis: Sole Heir and… ability appears very peculiar. I realize that it's once per game, but again…too peculiar. As to Knowledge of the Occult, I like the fact that you're balancing the extra die conferred by the Clue token with some extra damage. This one appears fine.
Randolph: Gift of Prophecy is a peculiar ability, as more often than not you want the Investigator to actually enter, not avoid, the Gate. Knowledge of Antiquities appears to be a last ditch effort and may not be employed with any Combat but only if he's going to lose his last point of Sanity.
Overall, you have some interesting, albeit unbalanced, abilities. If you haven't had a chance, check out the hundreds…possibly thousands of created Investigators under the Fan Creations area.