Check with local authorities but in most locations, if the fence is within your property lines, you can erect whatever type of fence is permitted under the local regulations. In most cases, a 6' privacy fence in the backyard is perfectly acceptable. Since it is 100% on your property, no neighbor permission is required.

Treat the dog-safety issue and the ownership issue as two separate checks. Before removing anything, verify the boundary from a survey or reliable markers, confirm who owns each existing fence, get any demolition permission in writing, check height/setback/permit rules, and have underground utilities located. If space allows, the lowest-conflict layout is usually a new fence entirely on your side, with posts and footings inside the line and enough access to maintain both sides. Mark the proposed line with stakes and string first, then check gate swing, bin/mower access, drainage, sight lines from the patio, and whether a narrow trapped strip would be impossible to maintain. Solid panels can be concentrated where the dogs approach rather than automatically making every section identical. Until the permanent fence is approved and complete, use a temporary secondary barrier so the safety problem is not dependent on verbal permission or the construction schedule.

Keep the chain link fence and plant in front of it. It will be prettier and cheaper. If you want, put another fence in front of the split rail.

You can put up the privacy fence in the back yard, you can put up a privacy fence on the split rail side and you can put up a privacy fence on the neighbor's side, who doesn't want a privacy fence. You must to install it correctly. You must install the new fence entirely on your own land, follow your city's zoning bylaws regarding height and setbacks. In my town, the finished side of the fence must face the neighbors.

Add oak leaf hydrangeas, that grow very quickly, under plant with hostas and other shade plants for easy care gardens that will soften the fence look.

The chain link fence neighbor: she can keep her fence or remove it...it's her fence, her call.

If you don't feel the need for privacy from her, let it be and plant another style of garden on your property. Another thought is to ask her if the two of you could choose a beautiful climber for the chain link. You can offer to pay for it.

As for the dogs, I hope the neighbor who has been bitten twice and has gone to the police or the animal control officer. Her neighbor is responsible for all her medical bills.