The general pull rates for Pick Up and Ordinary Recruit banners are:

4% for SSR units
43% for SR units
53% for R units

The SSR rate are further divided into:

2% for the Pick Up banner unit
1.5% for non-Pilgrim SSRs in the standard pool
0.5% for Pilgrim SSRs in the Pilgrim pool

Note: Some banners, like the Tutorial and Social Points Banners, have different rates. Please check the respective banner simulator for details.

Yes, Goddess of Victory: Nikke has a pity system with two types of currencies: silver and gold mileage tickets.

You can exchange 200 gold mileage tickets for a unit from the current Pick Up banner.
200 silver mileage tickets can only be used for a spare body of a unit available in the Mileage Shop.

Gold mileage tickets can only be obtained by rolling in banners that allow the use of advanced recruitment tickets.
1 roll equals 1 gold mileage ticket.

Silver mileage tickets are obtained by rolling on the Ordinary Recruitment banner.
1 roll equals 1 silver mileage ticket.

No, they do not expire. You can save gold tickets for future banners and silver tickets for desired units in the Mileage Shop.

Most Pick Up banners last 2 weeks; special event banners can last up to 3 weeks. Check in-game newsletters for specifics.

Pull on banners that feature units you want or need. You should also consider alternative ways of recruitment like molds, mileage tickets, and potential re-runs. Some players roll for Waifus, while others focus on Meta. If you're unsure, it’s best to save your resources for a banner that interests you more.

TL;DR: Pull if Waifu or Meta.

Unlike other gacha games, Goddess of Victory: Nikke doesn’t offer incentives for rolling 10 times at once.
Some players prefer 1x rolls to save gems/tickets, for example, they might get their desired unit in less than 10 rolls.
Others opt for 10x rolls for because it's faster. Ultimately, do what works best for you.

Yes, you can get free pulls through the Social Points banner, with each roll costing 10 social points.
The pull rates for this banner are a bit different from the standard recruitment banner:

2% SSR
43% SR
55% R

Note: The Social Points banner does not include Pilgrim units.

Players earn 1 social point for each friend on their friends list, up to a maximum of 30 social points daily. This allows for 3 Social Points rolls each day.

It depends.
Pick Up characters are added to the gacha pools sometime after their banners end. However, limited-time units, such as Seasonal and Collab units, are not added to the gacha pools. Seasonal banners have had re-runs, allowing recruitment for those units, but no Collab event has had a re-run yet.

Please refer to the in-game newsletters to see if and when a unit will be added to the gacha pool.

No one knows. So far, no collab event has had a re-run.