Welcome to the forums. It's good you are thinking things through and writing out how you feel. I'll try to answer some of your questions.
Quote1. My feelings about transitioning are always fluctuating. Sometimes when I imagine myself in a newly transitioned male body, I feel ecstatic but other times I feel very scared and just have this sinking feeling. There's no doubt in my mind that I am a male (at least on the inside) but is it normal for emotions to be so bipolar about the idea of transitioning?
Being scared and unsure about all of this is very normal. I would just take it one step at a time. Small steps, even before you see a therapist. You could order a binder and start wearing it around if you haven't already. I don't know what you do now so my apologies if you do this stuff currently, but you could get a male haircut. If a binder and haircut seem like too much because you don't want your family to notice, get male cut jeans or a pack of boxers.
When I started taking small steps, each step I tried made me feel so much better. I was unsure at first, but as I kept moving along with small things like this I knew transitioning would be the absolute right decision. The first time I wore a binder was euphoric. You don't have to decide now - try things out, see how you feel. Trying to deal with it all at once is overwhelming, and by no means do you need to make the permanent decisions without testing things out a bit.
Quote3. Although I know I'm male and identify as one, sometimes it seems odd when I refer to myself with male pronouns or other terms (for example, it felt a little strange to call myself a "whipping boy" up there). When other people use male pronouns for me though, it feels good. Maybe it's just because my masculinity has always been criticized by my family but has anyone else ever experienced this?
It could be because of your family - it could also just be what you're used to hearing inside of your own head when you think about yourself. Since hearing it from others is very validating, maybe you just need time to get used to thinking of yourself as 'he' - the pronoun. I had similar weirdness and disconnection when I changed my name, but now it's natural.
Quote5. Are there support groups for FTMs with family problems and/or financial problems? (Because lucky me, I have both.)
Unfortunately, many trans people have family problems when coming out. Surgeries are very expensive, so also unfortunately - many of us can commiserate about financial issues.
Quote7. And one final set of questions. How difficult is it to remain stealth? After my transition is complete, I would prefer that no one outside of my immediate family and girlfriend's immediate family know that I'm not a cis male. Is it even possible to keep your previous gender undisclosed for things like jobs?
Being stealth takes some effort, and it requires cooperation from the people who do know. Some people move clear across the country to start over. Someone transitioning is something people always want to talk about for some reason, so it has a habit of making its way into new spaces. It is possible, however, but I don't have any experience to share on this one.
You don't have to tell your job about your previous gender - however, when you change your name, you may have to tell them the old name on a form, and you definitely will have to list it when filling out a background check. If you have a very obviously female name - don't stress too much over this though, background checks are normally completed by third party companies, and even if some random HR person saw it, there's a good chance they won't make the "he must be transgender" connection.
Don't be afraid to keep posting here, there will always be someone who is glad to answer - this site has been indispensable to me during my transition.