Hmm. Many of your questions are a 'how long is a piece of string' question, and others will depend very much on your dance school.
DD (now 10) has been dancing in festivals in group dances since she was 6, and as a solo dancer since just before her 8th birthday. Her dance school enters festivals 3 times a year, and DD is expected to be available for the group dances at every festival. However, the choice to enter as a soloist / in a duet is up to us - so although so far she has entered every one, that is not something we have to do.
DD takes tap, ballet and modern theatre lessons, and as a result has solos in ballet (her best discipline), character (danceable by a dancer who has had ballet lessons IYSWIM, it's not a separate discipline), modern theatre, lyrical (a sub-branch of modern theatre) and tap. She does not [yet] have a National solo, as that is something that her dance school does not focus on, nor a song and dance solo, as frankly her voice isn't up to much.
A child who just does ballet and tap could do ballet, tap and character. At DD's bdance school, there is no particular push to expand the range of lessons in order to expand the range of solos IYSWIM - many girls don't do tap, and don't have tap solos. At festivals, I see children who only do 1 type of solo - just tap, or just modern, or just ballet - all the way through to those who do vast numbers. There are some 'aggregate' awards, but usually adjudication is per class, so it doesn't matter how many solos your child does, they will be judged on their dancing in that class IYSWIM.
Schools vary in how solos are taught. DD has weekly solo lessons year-round, where she learns new solos, practises old ones, and runs through anything else she needs to - so when she was in panto last year, she caught up on a lot of syllabus that way as she was missing her normal classes, and this year she's preparing for the ISTD ballet awards. Other schools just teach a solo over short series of lessons, and those then cease until the next solo needs to be learned IYSWIM. There are probably other models of doing it - perhaps learning solos in class time - but I haven't come across those.
It's expensive, yes. The parental taxi service is hard pressed on occasion. Luckily, as a teacher, all but 1 of the festivals we do are in half terms, and the third one is the most local so other mums often take and look after DD. DD is never going to be 'a dancer', and I have no ambitions for her to be so. However, she has admirable poise, self-discipline, work ethic and 'presence' which she has gained from dancing.