which camera is best for conducting the interview constantly as one-man show?
Let me start by saying that $1800 is a very, very small budget for this kind of a setup.  Bordering on impossible to get the kind of results you probably are hoping to get, if not over the line in to impossible.  

Video is sadly a rather expensive field to get in to and a professional quality camera the likes of what most popular YouTube shows are produced with runs more than your entire budget for one camera alone (often upwards of $5 or $6k per camera when including lenses and support/stabilization gear) and that's considered a fairly "budget" option by the wider video production industry (where broadcast and movie cameras are in the $60,000 or even $100,000+ range).  

Now, you don't need the same quality they are using and thankfully lower end gear has come a long way, but even minimal costs run close to or over your budget and it will show in the final product, just so you know what you are getting in to.  It will still be much better than Uncle Bob's family home videos, but it won't look like a top YouTube production either.

So my approach to answering the question is basically to give you what I think the bear minimum is to have any chance of achieving the quality you are hoping for.  It makes sacrifices in areas I'd personally rather avoid sacrificing in, but it's the only way to give you a chance of accomplishing your goal in your budget.  If you could increase the budget to $3000, it becomes far simpler (or considering used gear could also do it, though that really depends if you have a reliable vendor of used gear in your area.)

Personally, I'd still look at DSLRs, particularly Canon DSLRs with Magic Lantern support.  With a remote, it's not a larger distraction than having to stop for re-takes which you will also likely need to do for a top quality end product.  The way you do it is that you have two or more different cameras running and you start them on an offset.  As long as you check every few minutes, you can restart one at your leisure because the other is filming.  It isn't hard or even something that noticeable to your interviewee.

That said, the best bet is to have both the quality of a larger sensor and better optics crossed with being able to record without interruption.  Using a camcorder in your price range is going to have a noticeable negative impact on the footage, but using a Canon DSLR that Magic Lantern supports, it is possible to bypass the 30 minute limitation and have the camera automatically start recording a new file when it reaches 30 minutes.  There might be a gap of a second or two, but it won't matter since the other camera will have a shot then and you just edit around the gap.

Magic Lantern is technically a bit of a hack, but it is a well established third party "firmware" (it isn't actually firmware technically) that offers a wide range of enhancements for Canon DSLRs and makes them far more powerful video platforms.  Alternately, I believe there is also a similar hack for Panasonic's Lumix GH2.

Another decent option, though again limited, is to go with GoPro cameras.  They will get you good quality sensors and video but they limit you to the built in fixed lens.  As long as you don't mind positioning the cameras carefully though, they are cheap, light and high quality, and they don't have the 30 minute recording limit of DSLRs.  They won't get quite the same quality, but it's probably the best quality you'll get in a non-DSLR for your price range (or anywhere close to it).

Don't forget that you need tripods either.  At your budget, you won't be able to get tripods that are good for video, but since they are static shots, that shouldn't matter too much.  (A halfway decent video tripod is around $300 alone.)  For your purposes, go with the cheapest tripod that can hold your camera still, probably around $50 to $80 a piece.

As for lighting and mics, $1800 budget is really not a whole lot to go on, even for cameras alone if you want solid quality.  You are pretty much looking at getting either a stand alone recorder like a Zoom h4n or, if you can have a PC present, a USB audio interface that accepts XLR inputs (would be a bit cheaper than the h4n) and a couple of basic Lavilear mics (the kind that clip on to your shirt) or perhaps shotgun mics with stands depending on how quiet the room will be.  The Lav's are honestly probably the best bet since you don't have anyone to monitor the audio.

Lighting, you'll just need to use what little you have left to pick up some basic lights to fill out the kit.  We're probably talking home depot (hardware store) clip on light cheap with some home made diffusers and reflectors at that point because you will only have $100 to $150 left in your budget probably.  You might be able to squeeze in a really cheap quality lighting kit, but I don't have any direct experience using those to know if they would be substantially better than the home depot solution.

If you do have a reliable used gear vendor and don't mind going used, get used DSLRs (perhaps T2i cameras) and then use what that saves you to get a decent lighting kit and/or slightly higher end LAVs.