HIIT... expert advice please!

This is an interesting question and I am waiting for some answers from the big guns on this forum. But from my knowledge the great thing about HIIT is that you can freestyle it. There is no hard and fast rule when it comes to HIIT, the main principle is that you go as fast as you can, then slow it down. That's it. Anyone who says you HAVE to do, eg 30 seconds high, 60 seconds low, then they need a reality check (or at the very least provide some evidence for their claims).

Your HR seems fine. I don't think you should be overly concerned with what your HR should be. Rather, go purely on the time of high and low intensity. HIIT can increase your aerobic and anaerobic capacity and VO2 max (as well as EPOC compared to steady state cardio, see below). If you are resting primarily until your HR reaches, eg 135 bear in mind that as your fitness levels adapt to HIIT your recovery time will decrease, so you might start HIIT taking 3 minutes to reach 135 HR, but 4 weeks later you might only take 1 minute to reach down to 135 HR.

I would suggest doing cardio on your days off lifting to begin with. In your case, your lifting sessions are going to be the most important part of your training; lifting should take precedence over all other forms of exercise. The reason for this is that doing steady state cardio may affect your energy levels in the weight room. Having said that, if you do decide to do cardio every day and you feel absolutely able to complete your lifting session to your highest level, then by all means do cardio every day. If however, you feel as if your cardio is draining some of your energy (even partially) in the weight room, then back off the cardio a little. Remember, The New Rules for Lifting for Women puts a lot of focus on large compound exercises which use several muscle groups at a time and if you went for, eg a tough 5 mile run today and have your first session tomorrow morning, you might be very tired, other times you might not. Some days I have a weight session in the morning and a sprint or track session in the evening. Sometimes I have plenty of energy in my legs, other times I am completely and utterly shattered. Stress can also play a large factor in this - I'm sure looking after kids is a lot tougher than worrying about a couple of classes you haven't prepared for.

EPOC is excess post-exercise oxygen consumption - this creates an oxygen debt and a build up of lactic acid in your muscles, both of which must be eradicated once you've finished exercise. As your body restores this imbalance it increases your metabolism, so you burn more calories even at rest [from Nick Mitchell]. There was quite a good article on Charles Poliquin's website about this - http://www.charlespoliquin.com/ArticlesMultimedia/Articles/Article/854/Lose_Weight_For_Summer_Top_Five_Training_Tips_For_.aspx. It can also be extremely satisfying psychologically and therefore give your more motivation to workout as you feel better upon completion).

There are several ways to increase the difficulty.

Make the hard intervals longer (without going easier)
Make the rest intervals shorter
Add more repetitions

Any of those will give you the same result which is to spend more time at max effort.