Hi, I regularly fly DJI & Autel drones for construction projects as well as other inspections, outdoors and inside (use prop guards

).
Regarding airspace, assuming you are part 107, you would be good to go if flying in uncontrolled "G" airspace up to 400' AGL. Also if you are 107, you can fly up to 400' above a structure if you stay within 400' of that structure, and, do not go up into controlled airspace. Note that your altitude will also be limited by weather such as visibility and cloud ceilings, something that can be run into on long running projects.
If you are 107 and need to fly in controlled airspace simply use one of the LAANC enabled apps such as B4UFLY to get authorization up to published altitude. Otoh, if you need special authorizations to fly above published altitudes or in a NFZ or on a recurring basis, then simply go to FAA dronezone site and submit your request (plan ahead though as it can take days or longer sometimes).
Besides FAA 107 for airspace and aircraft registrations, most projects will also require drone insurance above and beyond your normal business CGL or other policies. Also some states also require a separate state license and registration per aircraft in addition to FAA/federal.
License, registration, insurance, airspace authorization and weather aside, construction sites and inspection locations can be challenging and fun at same time. Cranes are interesting to work around, likewise with buildings, finding location to fly from with good vantage points to maintain VLOS can be a puzzle sometimes. Sometimes there can also be other drones at the site so good to ask if there are any others known in the area.
As to permissions for occupied vs new built, there are two parts to that, one is where are you flying from, if from the construction site then you need the sites permissions and they may have additional safety requirements such as protective safety gear (glasses, hard hats, clothing, etc). For other buildings, assuming you can fly those without permission of the property owner specifically using their grounds for takeoff/flying from, bigger concern will be safety and privacy.
The denser the buildings, as well as with some towers, watch out for RF/radio interference.
Dronedeploy (DD) works good, their fly app can be buggy, as well as some functions only work on ios (e.g. 360 panos, mission chaining, etc) vs. the android version. You can also replicate a DD mission using a tool such as Litchi for your waypoints, pois, actions, etc including still photo, video in same flight, then upload to DD. You can also use autel explorer missions to fly perimeters or other missions, upload images to DD. For map/model/imaging/2D/3D, you could use DD fly app, pix4d, autel missions, etc and then upload to DD or pix4d or odm/webodm among others for backend processing. For 360 spherical, you can use the dd flyapp (ios), or builtin in for DJI, or Litchi, or Autel explorer among others.
Speaking of DD and Autel, currently the various fly apps (DD, litchi, etc) do not support autel from a mission flying standpoint which is too bad. Otoh, you can setup a mission using DD or Litchi among other tools and fly them across different DJI aircraft.
Speaking of DJI, some complain about the geofencing, however you can obtain unlocking license (free) for all DJI aircraft (from mini & FPV to mavics to phantoms to inspire etc). These get stored in your controller and aircraft, when arrive on a site, simply unlock your aircraft and away you go. To get an unlock code however you will need to provide DJI some documents such as any applicable COAs, waivers or other things to show you are flying safe.
Some wrap up tips for construction sites, inspections, stay heads up vs glued to your screen, maintain vlos, add extra strobe lights to aid with vlos, do a pre site inspection, drive or walk around to get to know site, whats new, whats changed, where are cranes, wires, things to watch out for, have a checklist and plan. Then have fun and fly the plan.