Can't do Squats or Lunges, help!

Do you get swelling on either side of your knees? If so, there may be issues with the Meniscus (cartiage between your femur and tibia - the 'cushioning') which will only show up on an MRI...
I had doctors tell me that there was nothing wrong with my knee, and after insisting on seeing an orthopaedic surgeon, found I had a completely torn ACL and torn meniscus in my left knee! Nothing wrong indeed...

Insist on seeing a specialist and having an MRI... The issues you are describing sound like it could be serious, and even I can still Squat and Lunge with the issues I have. (much to the disgrace of my physio!)
If you can't afford the fees of an Ortho, go and see a Physiotherapist, they may be able to pick up on something..

Have you tried the leg-press machine at a low-weight?
You can use less than your body weight, thus reducing the impact on the knee joint.
Use minimal weight to see if your knees will allow the movement pattern to begin with. If they don't allow, there is something not quite right happening in those joints!

Will I see fat loss from only doing the strength moves listed above?

fat loss comes from a caloric deficit

thats all it will ever come from

no particular exercise will change that

I have knee problems (patella tendonitis in both knees) currently and can't really squat, lunge or deadlift either. I'm trying to work around it a bit by doing the following exercises:

Single leg half squats
Half dead lifts (from a power rack at knee height)
Romanian dead lifts
Calf raises
Hip thrusters (these are new to my routine and are fantastic!!!)
Hip abductor machines
Plus some butt-strengthening pilates exercises - not weights but given to me by my physio

I know these don't really compare to being able to do heavy squats, lunges and dead lifts but it's going to have to do me for a while because I value my knees and want them at 100%. It's really hard to work the quads without bending the knees fully but the half squats sort of do it.

On that note - you probably should see someone and sort out what's actually going on with your knees

Also - rule no. 1 - any pain while doing these exercises and you sould stop

I hurt my knees last year over training and they have not recovered, even after going to several doctors. So, it hurts to stand up or sit down and prevents me from doing the best strength training exercises like deadlifts, squats and lunges. I know I need to lift heavy doing compound strength movements but I feel severely handicapped not being able to squats or lunge.

I'm currently doing overhead press, stiff leg deadlifts, bench press and rows; 3 days a week. 6-8 reps, 3 sets.

Will I see fat loss from only doing the strength moves listed above? Any suggestions on anything else to work my quads?

The type of compound moments you are doing are the reason why your knees hurt if you are lifting heavy. It is all dependent on your form and how much stress you put on your knees. You could potentially have chronic tendonitis or bursitis if you are constantly using your knees and considering your lifestyle and the way you are training, it is almost impossible for these problems to go away. In my orthopedic surgery rotation, I see patients with the same problems all the time and it could severely affect the rest of your life and cause future arthritis if you don't take the proper precautions right now. Message me if you would like some more information on this.

If you are trying to target fat loss, the only way to effectively do it is with a proper diet where you are in a caloric deficit. You can workout everyday of your life and still not see any weight loss if your calorie intake exceeds your daily recommended values. The best piece of advice I could give you on this is remember "diet is king and workout is queen". Hopefully this is just an acute stint of tendonitis that you can shake off with rest. Please be careful with the heavy lifting

I also have knee issues. I am a dancer and I do yoga, so between that and weight lifting (3 days a week) I am working on strengthening my muscles in ways that work for me and continuing to progress. I make sure everything I do leaves me with DOMs in my muscles and not joint pain (which is counter productive). I'm always looking for more info as well. Another one that might help you is step ups. Some people find they hurt their knees. For me it helps my knees. I make sure the step is just high enough that my knee is parallel (or lower), I hold heavy dumbbells and I step up, using my working leg (do all reps on one side and then reps on the other side), rest and then do more sets with rests in between. One point rows or one leg Romanian dead lifts are helpful for me as well. Sometimes using a machine or two can help you target the muscles while recovering (or the cords attached to your ankle to lift weights). I use the book "The New Rules of Lifting for Life" and find it extremely helpful.

I concur with physical therapist/specialist. Perhaps isometric type exercises might be useful without joint movement, but of course I can't know.