IPO / Capital Raising Nuances

I have some specific, detailed questions regarding IPOs that I’m hoping to get answered. I’m going to just weave them all into a paragraph. Feel free to answer in list form of in a narrative ... whatever is easiest. ******First off, in an IPO, what happens to existing, private shares? Are they “converted” into public shares? For example, if we have a $1bn company pre-IPO, and then it raises $250mm during the IPO, does that mean that now (forgetting about lockups), the company essentially has $1.25bn of stock that can be traded on public markets? Or, is only $250mm actually publicly available and the other $1bn needs to be sold off in the form of follow-ons? Is there a technical, official name for the actual conversion of private shares to public ones? ******Next, I know that under SEC rule 144 that there is about a 6-month lockup period before closely-held shares can be sold. If, using the example company above, would the company’s float only be around 20% (250mm of new shares / 1.25bn) immediately after IPO? Would the float % immediately jump up after 6-months and shares are no longer locked up? ******Finally, and I know this isn’t an IPO question but I thought I’d ask it here anyway, when looking at private capital raising, specifically pre-money / post-money calculations, let’s say (again) we have a company who’s pre-money valuation is $1bn. Let’s say they have a $250mm Series C round. According to a lot of sources I’ve read, their post-money would then be $1.25bn right? How come, then, Capital structure and financing is affecting valuation? I know that Modigliani-Miller tells us that valuation (in a simple world) is unaffected by how the company is financed. Are these valuations referring to just Equity Value (not Enterprise Value), which would mean that a $250mm influx of cash would change the equity value of the company? (If the answer to this question is yes then I understand fully no need for explanation). Thanks guys!