Why did my JPG come out bigger than the PNG?
Because the picture is the kind PNG is good at. PNG compresses runs of identical colour and keeps every edge exactly, which is what a screenshot, a logo or a diagram is made of. JPEG spends its bytes on smooth gradual change instead, so on flat material it uses more of them and blurs the letter edges while doing it. When that happens the honest answer is to keep the PNG, and the result says which file is which size.
What happens to the transparent parts?
They become the background colour you choose — white by default, or black. There is no third option that keeps them, because JPEG has no way to record transparency; a converter that appears to keep it has simply picked a colour without telling you. The result names the share of the picture that was transparent, so a screenshot with transparent corners is something you find out about here rather than after sending it.
Which quality should I choose?
Balanced for anything going in an email, a form or a web page — the loss is not visible at normal viewing size. Best when the picture will be printed or looked at closely. Small when a size limit is the whole problem, though if the limit is a specific number of megabytes, the image compressor is the better tool: it searches for the quality that fits under a size you name.
Can I go the other way, JPG to PNG?
You can, but it is almost never worth doing. PNG is lossless, so converting a JPEG to one cannot recover anything the JPEG discarded — it stores the already-damaged picture in a file several times the size. The one real reason is a program that refuses to open JPEGs at all.
Is my picture uploaded?
No. The decoding and the re-encoding both happen in this browser tab. The pictures people convert to send somewhere are often screenshots of private things — a bank page, a message thread, a form — and a converter that wants a copy in order to change a file extension is asking for far more than the job needs.