Most software has a way to use "velocity curves" to match your keyboard playing style to your synth's idea of what note velocity it wants. I don't use Logic so I can't tell you exactly where to find that option - it's probably somewhere in the MIDI setup dialogs.
But if you are literally not playing some notes at all, rather than just playing them too softly, you can't do much about that except play "louder". Similarly you can't do much about problem #2, unless you have an expensive keyboard with mechanical adjustments built into it - and if you had that, presumably you wouldn't be asking the question.
A simple spring-loaded MIDI keyboard will never feel like a real piano, but for some purposes you want "accuracy" from a MIDI keyboard rather than "musicality", and for that a simple keyboard can actually work better, so long as it's not so "heavy" to play that it is physically tiring.
You can get "semi-weighted" or "hammer-weighted" keyboards where the mechanism is closer to a real piano action. Some also emulate the "let-off" of a piano action, which allows repeated notes to play when the key is only partially released. As usual, you get what you pay for - and the top end of the price range is well over $1,000.
The ultimate solution is to use a real piano that is fitted with a MIDI output system - either a "ready-to-go" system like the Yamaha Disklavier, or you can get "MIDI output kits" to convert any piano to generate MIDI data. But a top-of-the-range Disklavier costs nearer $100,000 than $1,000!
Of course it's better to visit a big music store where you can try different makes to find one that feels right for you, rather than buying "blind" by mail order.