For the sake of simplicity (and because I'm familiar with it -- but bear in mind that I let my PADI membership lapse long before you even joined!) I'm going to assume you're talking about a PADI Divemaster certification here. The following applies equally to just about every other recreational dive training agency, as far as I know.
(PADI) diving certifications do not expire as such (although your first aid/CPR cert probably HAS expired, since those are usually only good for 2 years). So yes, you can certainly continue to use your DM C-card as proof that you are a certified diver, even if your (PADI) membership has expired.
The loss of your logbook may be problematic, as dive shops often want to see proof of the last time someone went diving before they are willing to rent gear to that person, or take them out to the more remote/adventurous sites. But this is a question of individual shop policy -- and some are more lenient/less money-grabbing than others**. If your old logbooks are gone, there's nothing you can do about that, but you can always start a new one, starting your dive-numbering at roughly the point where the last one left off (you must have done at least 60 dives, to get your DM cert). Get it signed off and/or stamped after your next dive, and go on from there.
As far as (PADI) membership renewal goes, this is really only necessary if you want to conduct (PADI's) official DM-led programs, or sign up for a (PADI) instructor course. Because you're not getting the standards updates, you're not technically qualified to do either of these things at the moment (you also won't be able to access the Pros section of the PADI website -- even if you could remember your login details!).
Not being a (PADI) member doesn't necessarily stop you from working as a diveguide and teaching assistant, though. In many places in the world you don't actually need to be a renewed (PADI) member to work as a diveguide -- and indeed many working diveguides don't even have a DM (or equivalent) qualification. The advantages of (PADI) membership in this instance are increased employability (the more professional dive centres and instructors generally prefer their guides and teaching assistants to be renewed -- and insured), and access to legal assistance and funding in the event of an accident (providing you can show that you were abiding by current industry standards!).
It's only a year or so since your membership expired, so unless (PADI) standards have changed drastically since I last taught, you shouldn't be required to do the DM course all over again before you can renew your membership. If you don't have access to a recent copy of the standards, you'll need to go along to your nearest dive centre, explain the situation, and ask them for the necessary membership application form, or get in touch with your local (PADI) office (see http://www.padi.com for contact information). Oh, and Put Another Dollar In, of course ;-)
All the best.
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**It was a fairly standard policy at the resorts where I worked that anyone who couldn't show proof of diving within the last 6 months was obliged to do a 'check-dive' with an instructor/diveguide before we would take tham out on the dayboats, or give them tanks to conduct their own dives. A check-dive usually cost about 5 bux more than a normal dive, some/all of which would go the guide as payment for the extra 'work'. But if a customer had a professional rating (DM or better), the check-dive obligation was usually waived (unless they REALLY wanted to do it!) ;-)