What is role of BW and frequency in internet speeds?
Are my assumptions correct? 


no, sorry.


  (BW) It is the amount of data that can be transferred simultaneously.just like more no. of highway lanes so that many cars can travel side by side simultaneously.


We need to talk about different meanings of the word "Bandwidth".


In networks technology, that often means "number of bits per second that this system transports". To avoid confusion, I call this "data rate" instead of bandwidth.I don't think your "highway number of lanes" analogy works very well, because it breaks down as soon as you start thinking about packets, latencies, congestion. I'd simply work with "number of bits per second", that is simple enough that it doesn't need any analogy, does it?
In signal theory, "bandwidth" means "the size of the frequency range occupied by a signal". 
For example, a FM radio station has a bandwidth of ca 75 kHz, and if it's at 100 MHz center frequency, that means it's occupying spectrum from 99.9625 MHz to 100.0375 MHz.Another example: human voice has an audio bandwidth of ca 4.5 kHz, it spans the range from ca 40 Hz to 4.5 kHz (rough numbers, really). 


Both definitions aren't interchangeable: For example, you can transport 1 MB/s over a 1 kHz wide channel, if your signal quality is good enough and you've got the right technology to make use of that, or you can at most transport 1 kB/s over a 400 MHz channel, if that channel is bad enough.


  It decides the speed of the data. So high frequency is like each car travelling at high speed.


That is totally wrong. I can't even slightly correct it to make it seem right.


  do higher frequency message signals require higher frequency carrier signals?


Higher frequency message signals typically mean "higher bandwidth message signal", and these need a carrier that is at lease high enough so none of the message signal mixed onto the carrier ends up at a negative frequency. That's all.


  So if we have both high BW and high frequency we can achieve higher data rates than either of them acting alone.


No, it's only the signal bandwidth together with the channel quality (especially: SNR) that matters for the achievable data rate. The whole point of modern communications technology is that the carrier frequency doesn't matter to the message signal.