I had a PC that was well above the recommended spec, but even that failed to cope. But I think a lot of it had to do with the software as much as anything.
As regards software, the software that comes with the TV card for the recording (provided it's not by Pinnacle - that's their recording software and not the editing suite as I have never used that).
For the burning and making your DVDs I would recommend either Nero 7 (although that can be difficult) and Adobe Photoshop Elements 3. To be honest this the hardest part, and I have had so many DVD fail to burn or in the case of Nero the programme crashes. Adobe is much more stable, but if you already have Nero I would make use of it's in built software.
In terms of syncing, the programme will do that all at the same time, it's just for one reason or another, the software hasn't been able to cope with full quality recording so the sound lags behind the picture.
I thought once that everything was fine (everything was in sync when watched on PowerDVD), but when watching the burned DVD, the sound was about a second behind the picture. So I would test your recordings in say Media Player before committing to disc.
Another thing to bare in mind is that often you have to pay around £5 for a license to be able to record in MPEG2 depending on the TV card bought
COST
That will totally depend on the unit you purchase, but you should get something reasonable for around £40. Most cards tend to be DVB-T compatible as well now so you can have Freeview on your PC. Have a look on eBay, but to be honest with all the hassle I have had in the past, I would suggesting using PC World's collect in store feature so the returns process is much easier if something goes wrong (and then you also benefit from lower prices).