I would say that this is highly dangerous. It is against US and Canadian code to not have outside ventilation for any fuel-burning appliance in your home; that's your furnace, HWH and stove/oven, assuming all are NG or propane. It is only acceptable to have a "filter-only" vent hood for your stove if it's all-electric (which BTW is the case for every single apartment I've ever rented; gas appliances may be cheaper on utility bills in the US, but a gas stove is a huge fire hazard and general liability for any landlord).
The code is in place for a very good reason; not only can inefficient burning of fossil fuels produce carbon monoxide and smoke (both of which continue to cause damage long after you've reached fresh air), but even when these fuels burn ideally, they remove oxygen from the air and replace it with CO2. CO2 in itself is not toxic in the same way CO and soot smoke are; as soon as you reach fresh air the symptoms of CO2 asphyxia begin to dissipate, while soot and CO poisoning ("smoke inhalation") can kill you hours after you reach fresh air. However, the consumption of oxygen and production of CO2 in a space with inadequate ventilation is a double-whammy for anyone in the same space; the oxygen is being consumed so there's less of it even in upper strata of the room's air, and as the CO2 builds it settles downward in a "blanket", pushing oxygen up towards the ceiling and away from you.
As the CO2 level builds, your body's natural "inhale/exhale" reflexes go haywire in a Catch-22 condition called hypercapnia; your natural breathing while in a high-CO2 atmosphere actually increases the CO2 levels in your blood, but the only thing your body can do to reduce CO2 levels is breathe. So, you start hyperventilating, which only exacerbates the problem. Should you pass out from lack of oxygen, you will not wake up if someone else doesn't get you out of the room or get some ventilation of fresh air through it.
Your landlord is running illegal housing. However, he may not know it, so be nice at first. Follow standard procedure for maintenance requests, and ask the landlord to install a proper outside vent line for this fume hood. If he refuses or drags his feet, you can call in the city's Code Compliance officials, or federal HUD (Housing and Urban Development) representative, and they will MAKE the landlord comply. Depending on the terms of your contract, the landlord may be giving you a free out by not living up to his end, meaning you may be able to break the lease at no cost if this has gone on for some time with the landlord's knowledge and inaction.
Understand that the cheapest way for your landlord to fix the problem with tenants still occupying the units may well be to cap off the gas feed and replace all the gas cooktops with the cheapest electric setups he can find. If this was the reason you moved in, and you don't get a "free out" from this debacle, you may find yourself stuck with a spiral-coil POS.
Go the stove manufacturers' websites and read the installation manuals for the stoves, checking particularly for installation clearances.
You could also replace the drywall behind the stove with 5/8 or 1/2" (depending what the surrounding drywall is) tile-backer (cement-board) and/or cover it with 1/4, 1/2 or 5/8" spaced off the wall surface by 1/2" to 1" (look up "woodstove wall-protectors" for the idea) and cover with aluminum foil if going for the last bit of effect. Aluminum sheet (flashing material) spaced off the wall has a similar effect (radiant barrier.)
You may well have damaged (or missing) insulation in the back of your current stove if it's heating the wall that strongly.
Best Answer
As far as the smell your installer is correct, the paints and cleaners on the stove need to out gas. This can take a few hours to a few days depending on how hot you get the stove. When we used to install stoves we use to warn folks not to get the stacks above 500 deg and would leave a magnetic temp gauge on the stack. For inserts we recommended a bulb type gauge that could be placed on the stack and the gauge could be routed to the front. Getting a rip roaring fire can cause damage to the stack, a standard fire running for several hours can and will make most brick work warm to the touch and depending on the fans or just plain heat coming from the stove the mantle will heat up. Since you now have cracks the system should be inspected. I hope they added a liner (required in my area) if a metal liner was added and you did not melt it down it should be safe to use. I would get a mechanical bulb type temp gauge that has a range at least to 500f. And install that so you don't overheat the stack again. As for other cracks and issues we would need photos to be of any help there.