I have tried https://pollanimal.com/ and it's good. You can create your survey easily. It's good for students, universities and businesses too so i hope you will find it good enough.
Hmm... if you try to enter all of the percentage variables into the same model (e.g., MANOVA), multicollinearity might be a problem since the categories are related. If you transform to control for normality, you may also have to take additional steps to control for multicollinearity. The more you tweak the model, the more difficult the output will be to interpret and communicate.
Your last bullet is probably the best approach: reframe the research question a bit. You might try to make qualitatively meaningful respondent groupings based on the percentages in the categories. For example...
The distribution and quantiles of scores might help to guide this process, and you can limit the number of groups to avoid sparse cells. When everyone is grouped, it may be as simple as doing a Chi^2 analysis and some pairwise comparisons. Just remember to correct for multiple comparisons using Bonferroni or similar method.