You should get into a gym. Something is always going to be better than nothing. Not trying to insult, but even if you are a "total natural", a BJJ white belt or someone in a MMA class for a month can probably sub you. Why learn something the hard way when someone is already offering it to you? Why bleed and sweat when someone who has already done their bleeding and sweating is standing their with their hands out, saying, "here it is, here's what you need to know".
Sure, some people have learned some stuff on their own in the past. But I don't doubt it was refined by someone in the know later. You can learn something on your own, for sure. But going to the gym is just so much more effective I think.
Anyway. Into your routine: You can bench without a bench, you know? Back in the day, the concept and use of an actual bench meant for lifting, just wasn't there. So you can lay on a regular bench (like, for sitting on, y'know?) or you can lay on the floor. Just make sure your spotter is hardcore on the ball, and be wise with your lifting. Try to push it as much as you can, but not to an endangering point, not to the point where you'll drop the bar on your face.
I suggest lifting in one of two ways.
1) Do three (3) sets of eight(8) to twelve(12) of a higher weight than what you're doing now.
2) Do a pyramid style lift pattern. This is what I very sincerely prefer, but I do mix it up with other lifting styles from time to time. A pyramid style lifting pattern means starting with maybe ten reps of a somewhat light weight. Then, drop down to seven or eight reps, but of a heavier weight. Then, drop down to five or six reps but of an even heavier weight. I normally continue this until I hit one or two reps at my "maximum" weight, then I go back down a couple steps on the pyramid as cooldown reps.
^The way you're lifting now seems to me that it's more oriented far towards endurance than gains in strength. I suppose if endurance is what you're actually aiming for, you're successful to some degree, but I really don't lift for endurance. I leave my endurance and conditioning to running, rolling, and bag/pads.
PS. Add some running to your routine on the days you don't lift. It'd be even better to run a bit every day, but do the most of your hard road work on the days you don't lift.
I never use protein supplements. I agree and do the same basically as ahartleyvu. If you're doing serious strength training, you're supposed to be taking in about 1g of protein per pound of body weight, or so. All you really need to do is eat a little more meat to meet (hah!) your protein requirements at this stage.
There's some lifts you're missing. I'm not sure what you mean by "standing, lift to chest", but if that's not a deadlift, you need to add deadlifts. I'm a ridiculously hardcore proponent of deadlifting because of how many muscles it hits. The other big compound lifts are of course bench press, and then squat. Squating without a squat rack could be difficult, but just make sure your spotter is really on his game and/or maybe use a front squat variation. Those are, I'd say, the big three in terms of compound lifts that are almost mandatory.
Below is a link to the one of the main power and strength training threads here. Definitely worth checking out as it provides a lot more detail than what I've put here, and can probably hammer out a number of further questions you might have in terms of lifting.
Hope this helped a bit.
Edit!: By the way, pumping out those sit ups is awesome. But I'd say switch it up a couple times a week to add variety to your ab/core routine. Do weighted situps for example. Or hang from your pullup bar (or a jungle gym at a playground) and do knee raises. Or do leg throws with a partner. Things like that, just to keep it mixed up a bit.