Hi vegemite, congratulations on your BFP!
When I found out I was pregnant I already had a personal trainer a couple of times a week and was doing some regular yoga. I had conflicting info from each about what I should and shouldn't be doing. The frustrating thing is that unless your gym instructor really knows why/what they are talking about wrt pregnancy fitness you can be misled. My trainer was great but didn't give me the correct advice to be honest, he just didn't have experience of pregnancy fitness or the reasons why you avoid certain things. You'll probably find that your doctor will give you some generic info to 'take it easy, do gentle exercise and don't push yourself but otherwise stay fit.' I found most healthcare pros were pretty chilled about what I did and didn't offer any practical advice.
So, basically it's important to listen to your body, as you know. You'll likely feel more tired in your first tri and have more energy in your second. The things you need to be careful of are core muscles because of the way your stomach muscles split to make room for your growing uterus and all your internal organs shift about. If you overwork / strengthen your core you may have issues post baby when you have rock solid muscles in the wrong place and big hole in the middle! Anything to do with twisting, core or ab work is something you need to take pro advice on. Once you get huge (which is not really going to be until you are well into your 3rd tri) you will be more than aware of what you can do and how your body responds. Also do a bit of research about how relaxin affects your body. You'll likely get super supple and bendy because of it but this is not a reason to push your stretches further than usual!
I would see if you can get a one off specific pregnancy 1:1 session in your gym from someone who really understands pregnancy fitness so you can personally adapt or go easy on certain things in all your specific classes. It's probably easier than outing yourself (your instructor probably won't be so subtle in calling out 'you are pregnant so do it this way' lol - I got this in yoga all the time :-)
Oh and I'd avoid making up a different injury, you'll just get the wrong advice on what to adapt unfortunately - and it will be hard to cover your tracks when they start quizzing you about specifics of your knee strain!
I had no issues lying on my back until about 35/36 weeks. Pregnancy yoga says go for it unless it gives your reflux or makes you feel sick. You shouldn't sleep on your back for hours on end because it can hinder major blood supplies as baby2012 says, or be on your back doing lots of ab work, but I don't think there is anything inherently bad about just being on your back for a few mins at a time of exercises, if you are working your pelvic floor etc.
Don't know anything about heart rate, sorry! Overheating is easy to do (and quickly) as you have so much more blood in your system, and being hot and fainting is generally not fun so I think that's probably why you shouldn't push so hard you feel like you'll pass out! But good fitness and a moderately raised heart rate is healthy for you and baby to the level you are comfortable with.
Sorry, that turned into an essay 
Congrats and good luck for a lovely healthy pregnancy!
oh and finally, agree with misspolly re pregnancy yoga. I LOVE the preggie yoga and pilates classes I do but they aren't very much about exercise and really just birth prep, squats, relaxing, labour positions, it's very gentle and doesn't raise my heart rate at all. But by the time you are feeling huge with your resident you will be quite happy to lay on the floor for an hour!!