Having your data tier enforce data consistency through constraints helps to ensure your data remains consistent and provides cheap runtime bug checking within your application.
If you think constraints are not worthwhile you either have a small/non mission critical system or you are passing up a huge opportunity to improve the quality of your system. This cannot be understated.
Choices include: choosing a different RDBMS, reinvent your own metadata system or manually manage constraints. Manual management in queries without a metadata system quickly becomes infeasible to maintain and audit properly as schema/system complexity grows and unecessarily complicates an evolving schema.
My recommendation is to choose a different RDBMS.
Consistency checking is much harder than you may think. For example MySQL uses transactional read consistency which means the values you are checking against may not be the same values in the scope of another transaction. Consistency scemantics for concurrent access are very very difficult to get right if not bound directly to the data tier.
When all is said and done, even with a modest amount of effort put into manual checking, the likely outcome is that one would still be able to drive a truck through the corner cases you have failed to consider or committed an error in forming.
On your NOT NULL question... The obvious data field requirements are a good starting point. Here are a couple of other things to consider when defining column nullability.
It provides a guarantee that can very helpful when writing queries. Various joins may use NULL conditions to show a non-match of a table row separate from a NULL value that cannot be assumed if the condition allows nulls. (If NULLs are allowed, a match can mean either the row did not match or the row did match but the column value is null.)
The use of NOT NULL also helps define the rules for simpler queries matching values. Since you cannot say "WHEN value1 = value2" if both value1 and value2 are NULL the result of the evaluation is still false.