I've presently started doing strength training exercises - I start with 5 counts of stomach vacuums, then do 3 counts of planks, and then 20 pushups and then do about 7 or 8 other exercises, all utilizing 5 lb dumbbells, targeting the core. In all, I workout only for 20 mins. I am unable to do more than this at the moment, since my heart beats very rapidly and I sweat too much and am unable to carry on beyond the 20 min. I am wondering if this 20 min will yield me the weight loss and toning effects I am looking for... Or should I do more? (I also walk each day for 45 min on the treadmill but walking isn't challenging to me anymore, I can do it rather effortlessly). Ideas welcome.

you will get stronger and that is nothing but good.

weight loss comes from diet, and can be augmented with exercise, but the way to lose weight is by reducing your intake.

the greatest thing about strength training, other than ensuring a healthy life, is building muscle means you are increasing your basic metabolic rate. which is fantastic while trying to lose.

That is a great introduction to strength training!!! Way to go
I have a very similar exercise routine and have been at it for a month and a half and am already seeing improvements in the strength area. Like I have moved up a few pounds with the dumbbells and have added 10 seconds to my plank! I think you will see as you go on that the exercises you are doing now will get easier, like the walking did and you will be able to add a little more as you go on! Also, switching up the exercises you do is a good idea, this will confuse your muscles and push them to work harder. I find a lot of different ideas on pinterest, there are thousands of variations, some I can't do yet but some I can. I like to use the stability ball as well, there are many different ways to use that and it is great for your core. Keep at it and it will pay off

I agree, great start to weight training is using body weight and light weights. You'll know to progress, when it gets too easy you'll need to add weights and time and change up your workouts.

As for your cardio, as I said, when it gets to easy you need to change things up...increase the speed, incline, do high intensity interval training (HIIT) ... you need to change things up all the time to keep your body guessing...

Also you must eat clean, unprocessed foods... Eat as close to nature as possible...

Good luck let us know of you progress...