I'm one of those as well. I even did a little bit of opera once upon a time, and you're spot on. I remember that voice lesson where I sneezed and my teacher said "Like that!" It's all about air, air, air, and air. And the control of that air, and keeping your throat relaxed and singing all the way from your ankles. And then there's voice placement and stuff like that, but really, it's about air.
I don't recommend trying to strain yourself and using falsetto. Work with your own range and what's comfortable for you. Not every guy is a tenor, not every girl is a soprano.
I can't sing and play uke at the same time. It's too much like walking and chewing gum I guess.
I have the reverse problem of a crappy unreliable chest voice. I can't go low. Well, at least after the voice is warmed up (and talking during the day is enough for that), it's a crap shoot. I have a range down there, but when the vocal cords are tight, it tends to crack low and I loose all control of it. I've got a play coming up where I have to sing something in chest voice, and I've had to practice it two octaves up, just in case on the night with all those nerves, I don't have a chest voice.
The best kind of singing to my ears is what I think of as controlled speaking. Do what's natural for your voice. Stay in a range that's comfortable, and sing as if you're speaking and even if it's not perfect, if you mean it, then it will be music.
If you want to really push that throat, I'd recommend getting with a good voice teacher. All that junk about what oils and fluids (other than water) that are good for singing is all just a bunch of hooey. They're placebos. You think it helps and so it does. I knew a soprano who ate chocolate before singing - something all of us avoided, but we all had our rituals.. .they are just that, rituals.
If sung all my life in choirs, classical and jazz. I was lucky enought to have received some excellent training in college too. For those who don't know how to sing and it doesn't come naturally, you need to start singing with people who do know how to sing. They will teach you how.
In a nutshell here is what you gotta do: To hit the high notes you have to tighten the belly. And thats about it. If you keep your abs rock hard, the pressure in you lungs will support your vocal chords so they don't get strained. You can't believe how complicated the human voice is, especially to sing nicely. You must be careful not to hurt yourself. In which case you may just have to do it the old fashioned way; seek instruction (via books or a live instructor) do the vocal exercises, and gradually extend your range. First though you have to find out if you are a tenor, baritone, or bass. If you are a bass you can just forget it, you were born to sing low. Baritones can sing high but it takes alot of time and will almost always be the falsetto voice. Tenors, you wouldn't be asking this question if you naturally had a high voice, you'd be doin' it already. I hope this helps.