PNG to GIF

No. The encoder writes fully opaque frames, so transparent pixels are composited onto white before encoding. Even a GIF that did use its transparency feature could only store 1-bit alpha, meaning fully on or fully off, which cannot represent the soft anti-aliased edge a PNG carries.

That is the flattening. The anti-aliased pixels around the logo's edge were partly transparent; against white they became light-colored pixels. On a white page you would never notice. On any other background they read as a halo, and no post-processing can cleanly remove them.

Flatten the PNG onto the exact background color you will place the GIF on, using an image editor, before converting. If the background is not known in advance, GIF is the wrong format for the job and you should keep the PNG.

For a photograph, almost certainly not, and often the opposite. For a flat graphic with few colors it can be close, but PNG's DEFLATE plus filtering usually still wins, and PNG keeps the alpha. There is very rarely a size argument for GIF over PNG on a still image.

No. This encodes one still PNG into a single-frame GIF89a. It does not accept multiple files, and it writes no delay or loop blocks.

No. The PNG is decoded on a canvas, flattened onto white, quantised, LZW-compressed, and wrapped in GIF89a blocks entirely in your browser.