Good for your son's doc to think of Lyme! I haven't researched the possible Lyme/autism link, but have noticed commentary on the concept in various places. The sudden rise in autism cases and the rise in Lyme seems awfully coincidental .... looking forward to the research catching up on this situation.
To your questions (and I'm NOT medically trained, so fwiw):
1) can there be a non-IV treatment antibiotics--i dread the
idea of an IV for my child,
-- IV is usually used only when oral abx fail for whatever reason. Treatment failure on oral abx is less likely to happen with careful testing to pick up all the co-infections of Lyme (carried by the same ticks as Lyme), because they often need separate (different) abx from Lyme. So don't worry about IV yet. If your doc wants you to go straight to IV, do some reading and have a serious convo with the doc about wanting to avoid IV.
2) any issues with taking antibiotics while on ADHD
-- Dunno. Ask the doc.
3) are there homeopathic tx?
-- My Lyme doc was uninterested in vitamins and supplements, but they turned out to be critical to getting me well, on top of the abx. As to homeopathic treatments instead of abx, there are alternative approaches, but I don't know anyone personally who has gone that route. I did some reading on the topic but never went there. Stephen Buhner is a master herbalist who has published some excellent books on Lyme; you might look online for them. I ended up not going that route, but would certainly consider it.
Also, consider talking with your child's doc about the possibility of fungal/yeast infection in your child now or as a possible result of Lyme treatment; I don't know if there is a connection or suspected connection between ADHD/ADD/autism with fungal/yeast infection, but I have learned that there are those of us (incl. me) who are more susceptible to systemic yeast/fungal infections. Major amounts of abx for extended time can cause yeast problems for some of us (as it did for me), and whether that is also hooked into ADD etc. susceptibility, I don't know, but I have wondered. It can be managed, but it took my doc a ridiculously long time to realize I was having problems.
(Side note: to avoid systemic yeast infection from the long term antibiotics I was taking against Lyme and coinfections, my LLMD prescribed a kind of probiotic that is yeast-based, because it is not killed by the antibiotics ... only bacterial probiotics such as acidophilus are killed by antibiotics. Problem was that I am yeast/fungus-sensitive, and the yeast-based probiotic took over my system. My doc said he had never seen it happen before, but I did some research and it's not unheard of. Just fyi.)
Sorry if this is scary, but you'll be fine. It just means figuring out what works and what doesn't, and your son's doc sounds like a good one to have thought of testing for Lyme.
Best wishes, let us know how it goes --
PS about your son having no symptoms ... ADHD symptoms could be symptoms of Lyme; there's some thought that it could be, but I haven't read a lot about it.
... and kids are funny in how they show or don't show symptoms. I think their expectations of how they feel day to day are not as fixed as adults -- after all, a kid doesn't have a lot of years looking back to see what 'normal' feels like and compare.
I was very very ill with Lyme and a coinfection, and when I finally was diagnosed, I dragged my teen with me and insisted that testing be done, on the theory that we had been all the same places. Great resistance from said teen: "I feel FINE!" but testing showed Lyme and a co-infection like malaria -- the same infections I had. Much grumbling about taking treatment, but afterward, the comment was: "I can see how much better I feel now. I was tired all the time but didn't realize it or know why."
and PPS -- Lyme is everywhere. The docs have their little maps that show Lyme is rare, but the maps are based on standards of diagnosing Lyme that are very very high, meant for epidemiology and not for diagnostics.
Pets easily carry Lyme ticks, drop them in carpet and grass, and then hitch a ride home from the park or playground or front yard. Lyme and Lyme ticks are everywhere, just like mosquitoes.
Don't despair; the hardest part for most of us is finding a doc who 'believes' in Lyme -- and you have that!
My LLMD strongly believes there's a link between Lyme and autism. See has Lyme as do two of her children, one of whom has also been diagnosed as autistic. Google autism-in-the-christian-home, and, there is a book titled The Lyme-Autism Connection. I myself feel that I was infected at a young age and that it has had a profound effect on my development.