He needs an outlet, and dedicated time each day for a mix of that and training.
Tennis balls could well be your friend. Cover his eyes (a friend had a springerdor and used to get him to face her, then hold his head between her knees...) and chuck them into cover (long grass, brambles). Send him in to find them and bring them to you. Start it off simply, within a few feet of you - he should pick up the game quickly enough. You could even start in the house, if you could trust him not break things (including you!) in his enthusiasm.
Work up to more structured retrieving - maybe get a gundog dummy and watch a few videos.
While you're developing this, do a little bit of training every day - 10-30 minutes should get you a long way quite fast. Again, look for videos on sit, stay, stop, recall and heelwork. It also sounds as if he needs work on basic manners - not jumping up, staying on his bed etc etc. Reward him with praise/ food/ play if he gets it right. remove him from the situation if he's a prat - he will learn. Start with the easy stuff in less-distracting environments: gradually build up distraction, duration, distance and difficulty.
Once you can trust him to sit while you pop round behind some brambles and hide some dummies or tennis balls for him to find, you can really ramp up the fun and games. Play recall games, and self control games where he has to sit and wait before being sent for a treat he's seen, or that you know if there.
IME (we're on to our third and fourth working-line dogs), training and games wears them out more than just free running. Building him up to a 5 or 10 minute stay in the garden is going to knacker his little brain.
With the rabbits, either move them out of his way (I'm assuming you can't, or you would have done) or make it very clear to him that bothering them is an absolute NO. If he goes near them, he is reprimanded and removed.
If you genuinely haven't the time and mental energy to spend on him, it would be best to rehome him, sooner rather than later. You have a lot of dog there, who could be wonderful in the right home. Don't beat yourself up if that home isn't yours: by getting him into the right home, you are doing the best thing for him.
ETA: I blithely assumed he was working line from the behaviour. Also I second @Bupster who recommends the Ladies Working Dog Group - it's very friendly and helpful, and their basic guide to sit-stay let me finally show my younger dog (batshit, nervy and not very bright) what she needed to do (I'd taught the older dog, no issues).