Welcome to GBO, you have come to the right place.
Some times the plastic stocks shoot very well. I do not think it makes any difference on the stocks if they are wood or plastic. The difference is in the fore end. If there are accuracy issues caused by the fore end, you have a lot more options with wood, while with the synthetic you are pretty much stuck with it. I bought a synthetic stocked 44 Mag. I sent it in for two more barrels and a set of Cinnamon lams. I did not even try the plastic set. They may have shot well, but I did not even give them a chance, I just like the laminated wood better. I do not think the laminated stocks/fore ends shoot any better than the "pallet" wood ones, it just a personal preference. There are several good deer rounds that are available in the Handi. The 270 is a good choice as is the 280 and 30-06. I do not have any of those, but I do frequent here a lot and I really can not tell which is better, so I think it comes down to personal preference. It is just a personal preference again, but I like the long action cartridges in a Handi, because the single shot is not at a disadvantage with the longer case. If you open up your choices then there is the 308 and 7mm-08 which also have been reported as shooting good in a Handi. I do not think it makes much difference if the barrel is a regular or an ultra, but the Ultras do seem to be able to get decent groups a little easier than the regular barrels and the regular barrels a little easier than the Ultra Lites. The regular barrels are heavy, heavier than most other manufacturers regular barrels. Ultras or Bull barrels are even heavier, but not a lot. Some people look to save weight, some look for weight to tame recoil. That is personal preference again. An Ultra with a scope and laminated wood can weigh up to 9 pounds or more depending configuration and scope/mounts. Ultra Hunters are limited to 243, 25-06, 308 in deer calibers. Here is a link to H&R to look over what is available
http://www.hr1871.com/Firearms/Rifles/hunter.aspI am personally having more luck with mine lately for accuracy. They do shoot differently than a bolt gun and you have to get used to their quirks. They are out lined in the FAQ. A good scope is important too. To get consistent results you need to snap the action shut, cheap scopes seem to bounce their zero around with this vigorous shutting of the action. This gives the impression that the gun is inaccurate when in fact the zero is changing from shot to shot. There are many that have spent $200 for the rifle and $300 or more on a scope. I have been hanging around here for a while and it seems to me that the ones that are getting consistent results from their Handis across their whole collection do not use cheap scopes. It is just an observation I have made along with my own experiences. So if you get a Handi, do your self a favor; put a good quality scope on it, it may save you a lot of frustration. Good Luck and Good Shooting
