Consider some of the ultra-compact poles made specifically for travel, they come apart to make them smaller and may fit inside your backpack for transit on an airplane. Here is a link to LEKI but there are other brands offering similar products
http://shop.leki.com/en/category.php?product_category_id=8423
I like LEKI poles personally, they have a 7 degree angle on the grips and with the arthritis in my hands/wrists/arms I also use their anti-shock poles, which I really find useful on very hard surfaces like roadways, but it offers nothing on grassy fields, soft ground, etc. Do a couple searches and you will find some pretty extensive posts about trekking poles with photos. Sadly LEKI doesn't make a travel style pole with anti-shock.
As for grips, yes they are specific for Left & Right hands and yes there are Men's and Ladies grips
(mens grips are usually larger) and most people tend to prefer cork or cork composite grips. They work well in all weather
(wet/cold/hot/snow) conditions and your hands don't end up smelling like rubber/plastic. The difference is not so much the feel of the cork when you intially grip it but rather the overall way cork responds to the weather and tends to not become as hard in the cold, as hot in the heat, etc.
BUT no matter what brand you buy, make sure to buy the optional RUBBER FEET/TIPS/TIP COVERS so you don't make that clack-clack-clacking noise.