That looks to be a slightly modified Stevens #404 "Semi-Military" position .22 RF target rifle.  Somebody modified the finger lever for better offhand support and maybe modified or replaced the buttplate.  The workmanship of the modifications is quite good.

You can see the holes for the original Lyman receiver sight on the right side of the frame.  A lot of shooters removed this and replaced it with a tang sight or a scope.

That model is pretty uncommon, especially in the nice condition yours is in.  Most people who wanted a Stevens .22 rifle for military type shooting bought the #414, built on the Stevens 44 action.  You ought to order a separate centerfire breechblock and extractor so you can change calibers without doing anything irrevocable to the original parts.

The screw through the center of the receiver gave a surface for the spring ejector on the .22 RF extractor to kick off from,  It does not generally show up on 44-1/2 centerfire receivers.

The 44 1/2 action will handle centerfire chamberings safely but I am not sure about the .30-30. CPA will be able to tell you if it is safe. If it is, and CPA does the work, you should be able to do it without doing any permanent damage to a beautiful family gun. That being said, there is a lot to be said for leaving it exactly as it is and continuing to use it as your great grandfather did as a great .22 target rifle. They are extremely rare and the family connection makes it all that much better. I recently finaly obtained one of these and have not had time to realy sort it out and shoot it other than function. Beautiful family heirloom!
Just remembered that I have a 44 1/2 in .38-55 which was a factroy offering. This may be a better choice than the .30-30 if you decide to convert it to centerfire.

You can get "almost" 30/30 performance from the 32/40 using jacketed bullets and higher than cast pressures but still mild loads.  32/40 was factory chambered for the 44 1/2 I don't think 30/30 ever was

Boats

And John Boy - I'm prone to changing my mind.  Can we put together a comparison between the .32-40 and .38-55 and .30-30 for deer hunting and Schuetzen matches?  I know that a 200 gr. .35 cal cast boolit is hell on deer out to 175 yds, starting at 1700 fps from a .357 Maximum from personal experience handgun hunting in Iowa.  Does the .38-55 or .32-40 provide comparable terminal performance?  The Krag is larger than I'll go, so I'll say my choices are down to .30-30, .38-55, or .32-40.  I cast boolits, so jacketed is not required, though I don't mind the option being available.  Let's have it guys, I'm open to opinions and experiences.

Paul knows his stuff and if he says so no doubt the action would handle 30/30

Thing that I would think about is the Stevens Factory chambered for 32/40 never Chambered for 30/30  32/40 is in keeping with the original specs and would be a better selling point if the rifle is ever sold, and they all get sold sometime. Most of us on this forum would not pick up a 44 1/2 chambered in 30/30 off a gun show table. We are Stevens snobs true.

If you look at the Cartridge particulars in Col Whelans 1920 The American Rifle,  Jacketed bullets both, 30/30 170 gets 2008 FPS 32/40 170 is clocked at 2065 FPS. Little bit better B/C to the 30 cal but not enough any Deer would know the difference

32/40 it would be chambered in a traditional Stevens Caliber not a Winchester caliber

38/55 more capable than either and with the 38/55 you may be able to get away with Cast Bullets at more modest velocity and deliver adequate deer performance.  30's and 32's you need expanding jacketed bullets.  

38/55 will be much easier to shoot than 30/30 recoil and muzzle blast wise. I shoot both in competion Single Shot 38/55 Lever Silhouette 30/30 signifigant difference in how easy the 38 is to shoot compared to the 30.  38/55  does have higher mid range trajectory less sure hit distance than the 30/30. My 36A Marlin 30/30 24 inch barrel does not rise more than 2 inches zero to 150 yards.   Actually the trajectory is the 38/55's hold back not energy on target if heavy bullets are used. Mid range on the 38/55 is about 9 inches enough to shoot over his back if mis estimating distance.  38/55 is way more capable than any 357 magnum. Larger hole longer bullet higher S/D same velocity same mid range trajectory

It's all opinion though and that's mine

Boats

Your rear sight is a Pope design marketed by Stevens, 1900 to about 1915.  Less-than-marvelous reproductions sell for $300+ when you can find them.  Good reproductions sell for over $400.  You have an original that is probably original to the rifle.

They are a PITA to adjust for elevation, but once adjusted, they stay put.  Windage adjustment is easy.  I have one of the PU reproductions on my vintage Farrow gallery rifle.  It works fine, even if much sloppier than the original.

The sights were designed for lead bullet Schuetzens.  Mine can be adjusted for any range from 50 feet to 200 yards when shooting 22 LR target ammo.  After that, I run out of elevation screw.

I sent pics into Buffalo Arms.  They also ID'd it as an original Pope sight.  They wouldn't put a value on it, basically however much someone is willing to pay, as it is hard to find originals in good condition.  This one will stay with the rifle - the case coloring on the tang beneath the sight is 100% perfect, which tells me that it has been with the gun since my great-grandfather purchased it.  Though the offer I had is tempting - I could buy the repro that John Boy linked to plus another CPA barrel with what I was offered for the sight from a member here.