When you get off your flight at DOH, you will need to clear transfer security. In my experience, even at peak times, this is an efficient process. If you walk off the plane at T-minus 2.5 hours, you should be through security by T-minus 2 hours at the absolute latest. With that, you'd easily have an hour in the lounge, maybe even a little more.

The extra security for US bound flights is done at each individual gate at DOH. So, if you are in the queue for the screening on time, the plane is not going to leave without you. In my experience - in business class - it takes about 15 minutes to get through the process. First you queue for the security screening interview. While useless, it only takes a minute or two. Then you go through the actual security. All laptops and tablets come out (phones can stay in your bag). They usually get swabbed for explosives (takes a few minutes at most), and then you go through the metal detector. Shoes can stay on (at least if you have TSA PreCheck). In my experience, it's pretty painless. The choke point, if there is one, is the interview. Once through that, you usually can walk up to a screener (each gate has around 3-4 of them).

After you clear security, the gate agent "boards" you (i.e. scans your boarding pass) and you pass into a waiting area. Once actual boarding starts, you just show your boarding pass and you're on your way. There is a dedicated business class jet bridge, so they don't make you wait behind all the old folks in wheelchairs (of which there can be many). Be mindful there are no facilities (bathrooms) in the holding area.

I typically get to the gate at the hour mark and usually have a 10-15 minute wait before actual boarding starts. If it were me, I'd want to get to the gate no later than T-minus 50 minutes to scheduled departure time. Most (all?) US flights leave from the C Gates. It's about a 10-15 minute walk from the lounges and there are moving walkways. It's not that long.