2. I can safely tell you 90% of the time, your offer is based on your current salary + a minor increment and NOT the budget. There's only a handful instances in my career where a candidate is on 5K and the budget is 9-12K so the client offer 9K. Almost never happens because all companies have a dedicated compensation and benefit team and the recommended increment is always 10-20% (Malaysia is 15%). They will not do outliers like 40% unless this is a highly niche & critical role (a rubber R&D specialist with experience in nitrile glove). As far as I understand, it is illegal to ask for payslips and their current salary in US so many pay based on expected salary and if the candidate is able to demonstrate all the skills required say, for a L5 SWE role, his offer will fall into the L5 band..unfortunately, this is not practiced in both Malaysia and Singapore.  To your question, we do not tend to pay expected as well but try to bridge the gap between our offer and your expectation.

Unfortunately yes. The market is hot and competitive so it's pretty rare to see underpaid employees but sometimes these candidates are GOLD.
Negotiation is key - while most of the time, it's 20% max, I'd suggest you provide data to justify the steeper hike (Company leaned on higher bonus, lower base? Took a pay cut before to join start up? Company isn't doing well so did not get any increment for 2-3 years? You haven't moved for 5 years? Provide any new training/learning certifications to justify why you should be raised to a new bar. Be kind to the recruiter/HR person because they're your gate keeper to get the additional layer of approvals for you > global C&B team, HRD. Usually we are authorised to offer up to a certain band (anything beyond that will need additional approvals from global).

Should all these fail, don't give up just yet - ask for a sign on bonus or a pay raise upon confirmation. Never accept the offer the first time it's delivered to you (a smart recruiter will always allow room for negotiation). Jual mahal, ask for time to sit on it. Don't let them trick you " If you don't take this by tomorrow, the deal is off the table" - it's a LIE. No way they want to start the search all over again unless they have a strong second after you.

One final tip that works 90% of the time, have a competing offer/or simply let them know you're at final stage with 3 other companies - We freak out when candidates have multiple offers and most of the time, we will match whatever you have in the market (without bother checking your other offer letter).