Can pesticides penetrate into fruits and vegetables as well as on their surface? Is it possible to completely remove pesticide residue?

Pesticides can penetrate under the surface of fruits and vegetables with thin skin, such as apples, tomatoes and strawberries. However, they mostly retain on the outer layer of fruits with thick skin, such as bananas, oranges and avocados. Systemic pesticides, which are absorbed by the plant and distributed internally to protect it from pests, cannot be removed by washing. However,  washing can remove 20–80% of surface (contact) pesticides. Rubbing or brushing while washing and soaking in vinegar or baking soda solution can improve the process. Peeling effectively removes the surface residues. Also, cooking (heat) can degrade some pesticides. 

Yes they can specifically, systemic pesticides are designed to be absorbed by plant they circulate inside the plant's tissue protecting it from pests from the inside out.
Even non systemic pesticides can penetrate through the skin or enter via stomata

These types of question about pesticides, a generic category, are asking for answers about 100's of different structures with a myriad of physicochemical properties, phase transfer coefficients, attenuation rates, etc. and cannot be answered definitively until a specific chemical structure is identified.  Consider  the answer for any one specific pesticide active ingredient to the question from the perspective of environmental chemodynamics.  Those compounds with comparatively low water solubility (e.g., for pesticides that would be anything in the low ppm)  can diffuse more readily through the surfaces waxes and cuticles of plant leaves and fruit.  Compounds with higher water solubility, would not penetrate as easily and stay on the surface (but nothing is 100% this or that).  However, compounds of intermediate solubility can penetrate through the cuticle into the leaf matrix and than cross from one side of the leaf by movement in the cell walls, which are not barriers to movement of anything (in plant physiology this route is called the apoplastic transport pathway as opposed to the symplastic pathway that involves crossing plasma membranes).  Whether they can cross membranes or not depends on the physicochemical properties, but if they diffuse into the phloem, they are likely to move into the maturing fruit.  Those types of compounds will not be removed by washing but will be attenuated to some percentage by other processing methods.  If you use GOOGLE Scholar and the appropriate search terms, many tens of papers have addressed the issue.  Unfortunately, only until recently have the putative modern pesticides been examined.   In the U.S. for example, organophosphate insecticides are very infrequently found on fruits and vegetables because their use has been highly circumscribed since the aftermath of the Food Quality Protection Act (1996).  In the U.S. most pesticide residues detected on food are neonicotinoid insecticides and some of the newer fungicides (e.g., those in the strobilurin chemical class), but few contemporary studies have looked at these specific chemical classes.  I did find one that is freely available and includes one neonicotinoid and one strobilurin fungicide (all registered within the last two decades) (Zarębska, M., Hordyjewicz-Baran, Z., Wasilewski, T., Zajszły-Turko, E., & Stanek, N. (2022). A new LC-MS method for evaluating the efficacy of pesticide residue removal from fruit surfaces by washing agents. Processes, 10(4), 793.)  Most residue washing studies show partial removal, thus still leaving pesticide residues.  I could go on, but if you would take time to read regulatory documents regarding the risk assessments for the pesticides prior to commercialization, I hope you would see that any health concerns are overblown and without any toxicological support when a proper risk assessment is conducted.  Nevertheless, the best reason to wash fruit and veggies is removal of bacterial pathogens.  I wouldn't worry about pesticide residues but when I go to a grocery store and watch everyone picking through the commodity bins to find the perfectly looking apple, etc., I'm very motivated to wash the fruit to protect myself from everyone else's microbiome (of course I'm also guilty of picking just the perfect looking fruit).  P.S.:  A lot of conventional as well as organic growers will use inorganic sulfur compounds as fungicides, often forming a whitish sheen on fruit after spraying.  Inorganic sulfur compounds will not penetrate and most likely to be easily washed off compared to other types of organic compounds.  Be aware that on certain fruits, kaolinite clay preparations will be used to prevent sunburn and potentially achieve some insect repellent properties.  Sprays of these preparations, such as one commercial one called Surround, will leave a heavy white particulate.  In the U.S., packing houses will wash this off.)  

Pesticides often penetrate into the fruits. It is very difficult to remove all the pesticides from surface though through repeated washing maximum residue can be removed. For fruits it will be more beneficial to remove the pill else wash it with warm water before consumption.

Azadirachtin is a natural pesticide derived from the neem tree (Azadirachta indica). It's a key compound in neem-based products and is known for its insecticidal and anti-feeding properties. It has also heeling properties for skin diseases and its natural chew sticks  used in cure of teeth problems. there are many such natural compounds which can act as  pesticides it think may not effect or penetrate in fruits and also prevent exposure of  chemical pesticides to environment and health risk in humans. 

To me, use of herbal products as pesticides can resolve these kind of issues. They are safe and effective as well.