So Gatwick and Southend are both outside the London zone system, thus your tickets to/from those airports will be separate from your London journeys. Also, minor heads up (mostly important if you'll be arranging anything via email): Americans reverse their dates, so when you say 7/6-7/12, we read that as 7 June to 7 December.

Gatwick to Herne Hill you can buy at the airport, it looks like the shortest journeys run to Victoria and back out to Herne Hill, but it'll be a bit cheaper if you avoid buying/taking the Gatwick Express and stick to the Southern services that only take a smidge longer.

For 6 to 12 July, your travel would benefit from price capping if you have contactless credit or debit card. You don't have to do anything or buy anything in advance for this, just start using your contactless card on the Oyster readers when you travel and you'll automatically benefit from daily price capping based on the zones you use, and there are also Monday-to-Sunday price capping that would be perfect for your dates since they start on a Monday and end on a Sunday. If you don't have contactless you can also buy a 7-day travel card for the zones you need (Herne Hill is on the boundary of zones 2 and 3, which means it will act like either zone in whichever way benefits you–coming out to Herne Hill from central London would be a zone 1-2 fare, but coming in to Herne Hill from Croydon would be a zone 3-6 fare, for example). If you only plan to be travelling from Herne Hill to central London, a zone 1-2 seven day travelcard would be sufficient. A seven day travelcard would include National Rail trains within your travelcard's zones, London Overground and Underground services, plus buses (travelcards for any zone include London bus journeys to/from/within all zones). I would not recommend any kind of pass that combines transit and tourist stuff, as many tourist things are free anyway, and your time will be limited. I've only ever heard of these combo packs breaking even, I don't think they can really save you a ton of money unless you run yourself ragged going from tourist event to tourist event.

Assuming you don't use contactless, there are a couple of ways to get a 7-day travelcard. This is going to seem a bit weird, but if you bring a UK passport sized photo to Herne Hill station they will issue you with a 7-day travelcard on paper for £32.10. The advantage of this is that you can use the travelcard to get some deals on the Days Out Guide 2FOR1 offer. Check out this link for a list of places and offers. Now you'll need a paper travelcard to get these offers. The other way to get a 7-day travelcard is loaded onto an Oyster (which is our contactless transit payment card). An Oyster would also carry a £5 deposit, so the initial price would be higher, it doesn't get you the 2FOR1 deals, and it's not sold at your local Herne Hill train station. On the other hand, it doesn't require a photo, and if you're going to do this you may find it easiest to do it while you're at Victoria in between trains from Gatwick to Herne Hill since they are sold there, and breaks in journey are generally permitted for the types of tickets that are sold between Gatwick Airport and Herne Hill. Victoria also have automated photo booths for taking paper travelcard photos, so you might consider that too if you opt for the paper route (which I'd suggest). In any event, to buy the Oyster version, go into the Underground station and buy it there, to buy the paper version it'd probably be easiest and fastest to do that from Herne Hill station itself (once you have your pictures).

Assuming you're heading to Southend Airport, you'll really want to do that journey by train. Trains to Southend Airport run from Liverpool Street station, which isn't horribly convenient to Herne Hill. If you buy your ticket from Herne Hill station to Southend Airport station, it will be for a train from Herne Hill to Farringdon, with a tube transfer to Liverpool Street, then a direct train from Liverpool Street to Southend Airport. The tube transfer would be included in the ticket (be sure to collect your ticket when it's spit out of each ticket barrier you use!), but it would be a bit of lugging bags around stations.

I hope this is all correct as I'm running on about 4 hours of sleep and so am very groggy right now, if I've misstated anything I'm sure I'll be corrected Good luck on everything and let me know if I can offer more help. I live one neighbourhood over from Herne Hill so I do know the area a fair bit, PM me if you need suggestions on things to go / places to eat in the general area. For hopping around in the adjacent South London neighbourhoods, buses are generally easier and much more frequent than trains (i.e., if you wanted to go to Brixton just hop on a bus, included with your travelcard). Sorry for the lengthy post!