This depends on how important a core value this is in your home. There are a ton of options for modest swimwear, cute cover ups. I think your concerns are valid and you should present some alternatives and let your daughter choose from those.
My answer on this one is easy: Bikinis mean skin cancer. My kids wear "rash guards" or swim shirts with trunks or swim skirts. No bikinis, trunkinis, or even traditional one piece swim suits. I find it is just too hard to keep them covered in sun screen and to re-apply as much as is necessary, especially if they are swimming. Here's the best part-- there is no changing clothes necessary! Go straight from the pool to the tennis courts, playground, or restaurant. They are already dressed!
I will link to my article on tanning and skin cancer risks at the end of this article.
Do you think that will work when they older, though? The pressure, especially for girls, to wear what all the other girls are wearing at the pool really picks up in the tween years.
My almost 9-year-old daughter has started asking for different bathing suits. I let her pick out her top and bottom and a rash guard. I don't allow an exposed midriff. She's happy with this. We've talked about my history of skin cancer and I think that is very frightening to her.
We wear long sleeve rash guards, too.
Keep in mind that when you are the one buying the swimsuits, you retain veto power. What is your child going to do if you don't get her a bikini...boycott the pool? Unlikely.
We drive 22 hours to Rhode Island every summer-- so I've got loads of car-trip experience, opinions and advice! At the end of this chat I will link to my articles on keeping your sanity on family road trips, avoiding the ER on vacation, and motion sickness.
In short, though, try scheduling bathroom breaks. This is easier for older kids, over 5 years. Limit drinks in the car. Make sure everyone knows they have to go every time you stop.
For toddler potty emergencies, we keep pull-ups in the car.
Regarding motion sickness, I can empathize... I spend my entire childhood carsick in the back seat. So now I am very aggressive about treating motion sickness.
Start with basic measures such as deep breathing and singing. Avoid eating too much.
If that doesn't work treat with Dramamine or a prescription such as Zofran.
For children prone to motion sickness, we are more careful about what the child eats before and during the trip. Fast food, excessive sugar tends to make a child feels worse if they are already queasy. We also take breaks to walk around and get a little fresh air every couple of hours. Limit screens and reading because this can also make motion sickness worse. Don't eat very smelly foods in the car, either.
Some of your daughter's fears are justified. Children under ten years of age are at high risk for dog bites and even death from dog mauling. At the end of this chat I will link to my article about how to teach kids to be safe around dogs. Here are some of the highlights:

There are almost five million dog bites every year in the United Statesand nearly one million require medical attention, according to the Centers for Disease Control. There were 31 fatal dog attacks in the United Statesin 2011, including a 15-day-old infant. You can read their stories at www.dogsbite.org. Children under ten years of age are at particular risk of dog bite. Although any breed can bite, pit bulls, pit bull mixes, rotweilers, and wolf-mixes are consistently noted to be the most dangerous breeds in multiple studies. Most dogs bites are from dogs known to the victim, owned either by the victim’s family or neighbor.
Here are a few key things you can do to prevent dog bites:
1) Never, ever put an infant or toddler on the floor with a dog.
2) Be sure that dogs cannot access children, especially infants, while they are sleeping.
3) Even if you do not own a dog, teach children age-appropriate interactions with dogs. When teaching children how to approach a dog, always be sure a dog is leashed and under an adult’s full control.
4) Teach children never to put their face at a dog’s level.
5) Do not approach an unfamiliar dog, even if it looks friendly.
6) Do not run from a dog or scream.
7) Remain motionless (e.g., “be still like a tree”) when approached by an unfamiliar dog.
8) If knocked over by a dog, roll into a ball and lie still (e.g., “be still like a log”).
9) Do not play with a dog unless supervised by an adult.
10) Immediately report stray dogs or dogs displaying unusual behavior to an adult.
11) Avoid direct eye contact with a dog.
12) Do not disturb a dog that is sleeping, eating, or caring for puppies.
13) Do not pet a dog without allowing it to see and sniff you first.
I think helping familiarize a child with a dog you know well in the neighborhood before you leave on this trip might help ease some of the anxiety.
My colleague Dr. Kevin Barton also wrote an article about why dogs are good for kids. And I have an article about why dog exposure can reduce allergies. I will link to these at the end of this chat.
It's appropriate for kids to have a fear of a dog they don't know.
Every holiday I see kids in the ER who have dog bites. When people visit friends and family, their sweet dogs can feel threatened by unfamiliar children. Then the dogs bite. It's key to teach children how to act safely around dogs so they can warm up to each other. See my tips below.
I see this in my own home. I think a lot of children will snack out of boredom. I know adults (myself, included) who are guilty of this, as well.