Altough i'm Polish i'm a big fan of the Byzantines. I don't know why but i really like their troops, and their map position.

I usually use: 6 Biz. Guard Archers, 4 Dis. Latinkon, 2 Dis. Biz. Lancers or Varangian Guard, 4 Vards, General, 2-4 flanking Cav, and some ballistas (which have a very accurate shot ans i really love to use them)

I play really differnet then most of you. I tend to leave Anatolia and the muslim factions alone.
I would probably leave the pannisula altogether but Nicea is just too good money maker just to abbandon it early. So i keep it and take also Smyrna and make it a castle, then i take Rhodes and make it a city just like i do with Crete.
Then i make some nice stacks of navy and block the paths between asia and Constantinopole, which leaves me with just one west to look at.
I take the emperor and blitz Sophia, Bucarest and Iasi. Iasi i make a castle. The rest cities. It's hard at the beginning but it's managable.
Iasi as a castle with some river crossings near is a perfect place to defend my second eastern border.
Now it's time to take care of the western factions. At the beginning i make a marriage betwenn my princes and the Polish prince... right the Polish... why? cos Hungarians are doomed within the cople next turns. Bran and Budapest are taken quickly, and the faction becomes rebel. i change Budapest to castle if it's possible, if not i leave a large garrison there, this will be me NW border, the city produces little trade as the area has no resources so i concentrate on the military aspects here. Bran i would change to a city or leave it be as a secondary fortress for fast unit production. In the west the Venetians are knocking to you for quite some time by now, especially if you take Durazzo quick as i did.
The key here is their fortress Ragusa. Take it and then kill off their militia with Vards. I make Ragusa a city making Zagreb a Castle to complete my NW border. The last thing is to take Iraklion, but it's obvious.
By now i have like 200000 florins and sit secure in my territory. The alliance with the Poles is strong and it never broke. Especially if you take some minor rebel lands and give it to them.
If the HRE starts to bother you sack and pillage Vienna and then give it to the Pope or the Poles. If you have some princesses always marry them to Polish princes.
This should secure you in Europe forever.
What you do next is up to you. Usually i jump Sicily and the "italian" isles. Then go For Tunis and march throug the Moors.

As for the "hordes". If they go for Constantinopole you hace the passages blocked by navy so only Nicea and Smyrna are in danger. You can try to defend them or let them fall. Your total income wont suffer form it. If they go for the north steppes you have Ias supported by Bran and surrounded by rivers, should be managable to defend.

You should really ally with the Poles, you will have at least 2-3 common enemies. You could try alliance with the Pope especially after some tribute and giving him some lands. Will secure you from the crusades against you. I ally also with the Turks as i don't plan to invade their lands. They usually keep their word for several years.

As for Byz pros:
- great HtH and Missile Infantry
- great early navy
- decent to uber cavalry
- one of the best economic potential
- not boring to play :)

As for Byz cons:
- very weak spear units
- no good gunpowder units
- can't do Jihads or Crusades
- multiple enemies