Ships & Modules

I think you could, just keep your speed up, don't get webbed down and you'd be ok. Big thing with hubs and havens is that the rats are meaner then in like L4s. But because you will have sentries and drones, you could do them. If you get neuted, your AB or MWD won't last in a Vexor or Gila, and for those ships you need your speed. Might want to look into a Proteus drone fit also, might work well, not sure though.
The reason I preach a RS is because it has the best shield tank BS and down, is the ONLY ship besides carriers able to solo L5s, 6 of 10s and up, and rock officer spawns solo So Havens and Sancs aren't much issue. Also because it's passive neuting doesn't affect it at all. It is VERY skill heavy, but if you can work up to it is'nt a great ship. Justt adapted to what you are fighiting, it might need more shield power relays, or replace one or 2 large shield extenders for shield rechargers, get as much passive shield rep as possible while still doing pretty good DPS. The RS starts out with 12,750 shield HP, it's a beast. So play with her and see which will passive shield rep you better extenders or shield recharges.
Find what works best for you and go with it. I liek the RS for a number of reasons, but I also use a golem and tengu, just all depends. If you need anymore help just mail me I'll do what I can to help.

You need to drop drones as soon as you land and they must be set to aggressive. Once the rats start shooting you, your drones will automatically aggro onto a rat. Do NOT tell your drones directly to engage. You do NOT need to lock anything. This is the best chance to not have drone aggro without using EWAR on your ship.

Do NOT do the Havens with the Chemical Factory, the Popes will pull range and make your isk/tick horrible. Stick to the stargate haven (the ones that give you the pop up as you warp to them) as the popes generally get stuck and stay inside the roid, meaning your drones have to travel less distances and thus better isk/tick.

There is no reason to upgrade to a Gila from a Vexor Navy Issue. The Gila is a shield tanker, and therefore has a larger sig radius, making speed tanking harder. The Gila also does not have a velocity bonus to drones making heavies slower than the Navy Vexor so less isk/hour. Also because the Gila cannot speed tank as well as a Navy Vexor (10mn AB only), it needs to facetank more dps. If you decide to go with sentries on a Gila (as you should be as there is no velocity bonus to drones), you will need to have a tank of around 250dps in a haven (assuming you orbit your sentries) - this gimps your dps to the same as a Navy Vexor. TL:DR You are essentially paying double the price of a Navy Vexor for a ship that does the exact same job equally or in some situations does the job worse.

While I heartily agree on this point, the rattlesnake is not an optimal choice for running forsaken hubs and havens. You are spending in excess of 800mil isk on a ship that will get you 20mil/tick when you can do the sites at 15mil/tick in a Navy Vexor that costs less than one quarter the price to fit and fly. Of course you are protected against neuting by having a passive tank and protected from tracking disruption because you are using drones and missiles but this also applies to the Navy Vexor as no guns are used and if you stay outside 34km of the elite cruisers in the anoms you will not be neuted at all and will rarely dip into armor. TL:DR Rattlesnake gets you 5mil/tick more than a Navy Vexor for 4x the price.

@Thaddeus Eggeras
I will happily tell you that a Rattlesnake will not be able to solo a 10/10 Blood Raiders Naval Shipyard even with a max EHP and Passive Shield regen fit. There are 8 Stasis web towers in the site that web you to 4m/s and 13 Neut Tower III's that will alpha your capacitor (i.e. no MJD'ing out). Also as a side note if you do get your MJD off as soon as you land, you are still dead as the cycle will not finish before the citadel cruise missile hits you and while your MJD is cycling, you have a 150% modifier on your sig radius. TL:DR You get alpha'd by the 180k dmg citadel cruise missile due to it being a shield tanker with a huge sig radius (I've lost one, thats why I know \o/). The ONLY reliable way to complete this DED site is to use a tripple plate abaddon, 2 caboose guardians, and 3 more dps battleships, and no you cannot get carriers in - its gated.

Ogres are the best but Praetors work fine too