Links to posts of public channels: t.me/channel/123, t.me/s/channel/123 and the same on telegram.me. A link with ?single at the end takes one photo or video of an album. Links to private channels (t.me/c/...), invite links, links to comments (?comment=) and to live streams don't work.
No. Posts of public channels download without signing in: the site reads a post the way Telegram shows it on the web. Sign in with Telegram only to get a file or a post in your chat with the @CmdSaveBot bot, so that the bot knows whom to send it to. Signing in does not give the site access to your chats.
Yes. Paste the link to the album's post and the site shows all its photos and videos in order, each with its own download button and one to save them all.
The site takes photos from Telegram's web preview, which gives reduced copies, often 800 pixels wide. Sign in with Telegram and press To Telegram, and the @CmdSaveBot bot sends the post to your chat in full size.
The site doesn't download large videos, files and music. Such a post can come to Telegram: sign in with Telegram and press Get in Telegram, and the @CmdSaveBot bot sends it to your chat. Deleted posts and posts of private channels can't be downloaded, and paid posts open only to those who paid for them with Stars.
The bot downloads the post through Telegram itself and sends every photo, video and file of it in original quality. A post over 2 GB comes as a download link. Posts over 4 GB aren't sent from the site: you can download them in the bot itself. If the bot can't write to you, the page says what to press in Telegram.