what you budget?
how peopl can help yo if you didnt give what you budget?

-motorola atrix 
-samsung google nexus S
-HTC desire HD / incredible S / sensation(heard got some probs, manufacturing defects i think)
-LG optimus 2x(laggy on stock rom, custom rom ok)

dunno which 1 macho for u 

1 and 2 can be done by most if not all phones above 2.2

If you're using Gmail then the Gmail app would serve you well. Provided the mail is received when you're on an active internet connection, it can still be viewed even if you don't have one.

You can probably go for any high end phones as of now. Galaxy Note would be preferable if you don't mind the size and/or prefer it to be big.

SGS2 is not recommended unless you can find the non-g version.

thanks xDarkKnightx!

for point 1, need to install anything or its pre-installed inside phone? can you suggest best phone which is abt palm since (standard size not like tab) since i'm not into gadget much thus dunno which phone is good/not..

No problem Just here to help.

As for your question, I'm not too sure of that. Maybe someone else can clarify it for you.

Palm sized would probably be smaller than 4.3" and since you're so geared towards a HTC, I suggest you wait for the MWC as current offerings just don't seem decent for their price.

The Sensation is not a bad phone though but I heard there are some touchscreen issues. It's only 1.2k now though .

Also, about your statement towards the battery drain of a 4 core phone, I can assure you it won't be a big deal.

Most apps now aren't even capable of running on two threads what more four? And those cores which are not used will automatically power themselves down. As far as you are concerned though, it should consume more or less the same power.