You're right that Rental Houses see Peaks and Troughs in activity dependant on Season, but turnover of staff isn't something I'd particularly attach to that as well. If they've got a position, and you can prove you're the right guy for the job, then there's no reason they won't bring you in on the floor. The reason you want to be in there initially is to get to grips with all the Camera Gear they have from a technical stand-point. The jobs will then come when your superiors have got to know you and feel confident to recommend you out on a job, or you make an impression on a Crew that have come in for Tests/Check-out etc. It's a route that I didn't take, but still think about.
I don't know about Lime Pictures, but what Arri can do is eventually get you on their Crew Diary Service. Again, this isn't something I've experienced personally, but I know a DoP who is and this surely helps him in terms of Kudos and to some lengths steadiness of work. It depends on how you want to do it; if you want to be doing your job on set all the time, then it might not be the best option. As another person said, it's good enough to have a working relationship with the Houses, but not imperative to make your start there. However, I'll re-iterate that the one huge benefit of working in a House is that you can learn everything you need to know about a plethora of Cameras, Lenses, Grip Equipment etc. that will make you infinitley more confident in working with said equipment when the job comes.
In terms of getting on to Productions, there are again a variety of ways to get in. The major one is recommendation from previous colleagues, so take Adrian's advice on board and keep those contacts close. The other, which some might disagree with is yes, contacting the Production directly through whatever means (Listings etc.). You might also like a DoP's work and get in touch with them, offering your services. The same goes for your 1st/2nd AC. If you can get a personal rapport going, then I imagine it will be much easier to land the work.
Finally, I would like to give a word of warning. Be careful not to throw yourself out there for something you might not be prepared for. As a Focus Puller once told me, it takes years to build a reputation but can take seconds to destroy it. Don't state to be something you're not, as it will only bite you in the arse later, unless you're lucky!
Hope that helps.
Sam