I upgraded my mower with a car battery,mainly because I got tired of those small batteries not lasting very long.
I had plenty of room in the stock battery location.
As mentioned,it is still only a storage device and won't produce any more power.
Even with my electric clutch though I didn't need to recharge it very much.
Make sure you get a good quality battery,preferably a deep cycle as they are made to be discharged and recharged numerous times.
Stay away from the walmart crap,they aren't built well and discharging them usually dooms them.
I use duralast gold series from AZ in all my vehicles including the mower for a long time with excellent results.
I use a redtop optima in the mower now and have no issues.
Not sure how much your looking to spend on the lights,but autozone has some sweet HID lights for 4 wheelers on the shelf. Can't remember price but it's either $89 or $149.
Had my eye on them a couple years now,just waiting on them to get clearanced
Wish I could help on the throttle control,haven't found a good way myself.
The above advice about the lever on the steering wheel is about the best way i've seen to do it,mainly for the reasons listed about being steady/stable.
I've held back mainly because I like my cruise control,though a 4wheeler brake lever with the tab that locks it in place might be an effective cruise control.
I use mine to mow still,pulling a 60" finish mower on 2 acres. So losing cruise in my case would really suck.