Well, you've got a well selected team there, to start with. So that'll help.
With that team, your core weapons should be the Sniper's.... uh.... the big green beam, I always forget the name of it.... and the Brawler's Dissolver. If you have the Epic version of the Science Exo, it's chain-gun is another weapon to rely on, one of the best in the game.
The Assault is sort of okay as an all-around sort, but simply doesnt have the damage output that all of the others can manage. It's better than it used to be, but still not all that great. As tanking goes: Try not to have to do it. At all. Your goal should be to defeat all approaching bots before they can get into a position to fire at you. And yes, this includes things with very high range... use the terrain to your advantage to deal with them. The Assault is a decent tank if it becomes absolutely necessary, but the main way to avoid damage is stealth. Enemies cannot fire on you when you're stealthed, and this includes the turn that you activate it. They also wont follow you, since they cant sense where you are. An important fact to remember. If you're playing on Normal, it's okay to tank some hits, but if you're on higher difficulties, or planning on playing higher ones later, do your best to learn to avoid getting hit whatsoever. On higher difficulties, many enemies do extreme damage of the sort that even a buffed up Assault exo cannot withstand.
As equipment goes, with that group focus on increasing the damage and range of the Sniper's beam, the Brawler's dissolver, the Assault's grenade, and the chaingun for Science if you have it. Alternate weapons gotten via the Epic versions can be helpful as well, but are not as important as those. Range, and damage, you really need both of these.
Lion's Den missions really arent too much different than normal ones. Just your typical set of bots, with the difference being that you can see what they are going into it, and that destroying them has an effect later in the game.
Assassinations however can be nasty. Bosses are typically very powerful, particularly the Viper (I think that's what it's called), who has the ability to take 2, even 3 turns for every one of yours, as it takes more and more damage. All of the bosses though have very high health and attack power, and of course unique abilities too. And it wont just be one or two of them in a mission. I've seen missions with 12 of them. Though, I play on the high difficulties, so that may be a bit lower for lower difficulties, I'm not sure.
When dealing with any truly dangerous bot, be it a boss or something like a DoomBot, you really want to use more than just weaponry to deal with them. Viruses can help a ton, and are one of the most important things in the game. Either the Science or Sniper are very good for using viruses. But that's not all; you'll want turrets, definitely, as they can do heavy damage with many shots, so long as they arent exploded too fast (they do best in groups), and land mines, when placed right, can do tremendous damage if you lead a strong target over a long line of them. I usually have the Sniper do these things because it often works well to have that one be the one with the highest Trap level, a stat which is really, really important for whichever exo is using it. It's important for dealing with mines, electric floors, and most importantly, tripwires. Of course, if you dont keep that trap level high enough as you go through the game, your mines and turrets will be weak, and you wont be able to pass over dangerous floor tiles of any sort without setting them off. Viruses are my favorites though; even the strongest enemies cannot resist these. This includes bosses. A bot that is hit with viruses can be considered instantly defeated, and of course will then fight for you. Gotta be careful though, some bots have a really high cost in terms of virus points, and you also have to consider that a virused bot can wake up other bots by firing at or near them, so consider the situation, as well as the bot you're virusing, before choosing to do so. You also need stealth to use a virus. An absolute necessity, because there's no other way to get into range without getting attacked a bazillion times. Ideally, you want to lure bots to a point where you can move around a corner and then sit there, waiting for them to get just into their firing range when coming along the wall, and THEN hit stealth, walk to them (they should be about a tile or two away when you do this correctly), and THEN hit them with the virus. Some bots are very high priority for virusing, such as the ShieldBot. Do anything it takes to grab that guy. The buffing type bots, Command and Leader (I think those are the right names), will buff YOU if you grab them. They heavily buff health and shield power respectively (and can give you enough shields to make it actually matter), and more importantly, they wont buff enemies when virused. Masterminds, however, will not buff your guys. When grabbing them, use the virus, and then switch to each of your exos before the bot wanders out of range, so that all 4 of them recieve the buff.
Also, some enemies are VERY high priority targets that you should always try to destroy first. All of the buff bots are truly nasty to deal with, ShieldBots make all nearby bots invincible until it's destroyed, and things like DoomBots are just absolute terrors. Make sure you know what each different bot can do, and keep those traits in mind, doing your best to exploit them.
Finally, you really absolutely have to make sure you dont pull too many bots at once. The whistle that you have, usable by all exos, is probably your most important tool. You should never be catching bot attention by walking in their line of sight; the whistle should ALWAYS be the only way that you wake them and call them to you. Them coming to you is advantageous; whereas you coming to them can mean death. Pulling only one or two or so is okay; keep that number as small as it can be. Again, not always doable, but try your best to make it happen. Holding the cursor over the whistle button shows it's range for the exo you're using, so check each exo and see who can get into the best position for safely pulling bots. Often, the science exo is not the one to use.
And of course, if you have done all 4 of the Bahamut missions and thus have all four Epic versions, dealing with these missions, particularly the assassin ones, will be much less risky. MUCH less risky.
As equipment goes, weapons come first; attack power, then range, then ammo. AoE for explosive weapons (including the dissolver) is also important. Computer stats are top priority though for the Science exo, and trap skill, while not as important as weaponry, is needed for whichever Exo gets to do the trap placing.
Beyond that, it's up to you. Probably the thing that's needed the least is shields; again, you should be doing your best to not get hit in the first place, and once you're doing high difficulties, shields will rarely be able to protect you whatsoever.
It is best to not do either of these two mission types early in the game.
And the final battle wont be too bad if you do a few of these over the course of the game. It's actually split into multiple fights, not everything all at the same time.
That's about the best tips for you, hope that helps.