Stratification should help, but without either auxiliary data for regression, or probability sampling, or both, you still run a substantial risk of unknown bias.
I guess that both probability sampling methods and non-probability ones can be combined in stratified sampling as described. However the analysis will have to be adapted to suit the non-probability convenience method adopted.
Yes, if there is no other option. At least you would have stratified first. Just remember that convenience sampling has the possibility of sampling error and lacks true representation of a population :0
Yes you can use convenience sampling technique after stratification, if you don't have other options. People in MR Industry are using it very frequently.
But If you want an ideal case(sometimes free of statistical errors), you should go with pure random/probabilistic sampling.
You are never free of uncertainty in a statistical/survey sample. With randomization and/or 'prediction' (regression), you can estimate variance, but bias is a more nebulous consideration. Testing can help. Post hoc bias assessment may help. But there are uncertainty issues with data collection/measurement which impact a sample, and even a census. Sometimes a sample may even provide data for inferences more accurate than a census when there are data collection and related issues. So, though it is best to try to assess uncertainty, there is always some uncertainty even about that.
I assume we are talking about quantitative data. Following is a link to an interesting and entertaining, freely available paper by Ken Brewer on the various methods, primarily for continuous survey data:
Ken Brewer's Waksberg Award article:
Brewer, K.R.W. (2014), “Three controversies in the history of survey sampling,” Survey Methodology,
(December 2013/January 2014), Vol 39, No 2, pp. 249-262. Statistics Canada, Catalogue No. 12-001-X.
What exactly do you mean by convenience sampling (this covers a lot of evils)? The key point is the representatives of the units you are given. Most convenience samples are biased in both the statistical & non-statistical sense.Thus to extrapolate them to the population you probably need an external source (census?) to calibrate & balance the sample so it is representative of the true population. Stratifying may help you calibrate better but it could make things worse too. Theoretically your stratification variables should cover all possible explanatory stratum variables not just the ones supplied with the convenience sample. If the convenience sample does not include units from all possible explanatory stratum then it may be valueless. For instance if the convenience sample only covers people ages 25-45, then you will not be able to estimate characteristics for the population outside this age range. Sometimes you have no choice but to use a convenience sample and a lot of people do what you are proposing but that does not make it right. A lot of well-known & respected statisticians consider convenience samples to be valueless. Think carefully about the potential biases in your sample & if your stratification variables cover all explanatory variables before you go ahead. Fundamentally, you are modelling not random sampling so make sure your model is reasonable.